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OPINION McKEAGUE, Circuit Judge. This is an appeal from a denial of habeas relief. Petitioner Stanley Jalowiec was convicted of murder and sentenced to death. After the Ohio courts denied Jalowiec’s requests for appellate and post-conviction relief, the district court denied all forty-seven claims of error asserted in Jalowiec’s petition for writ of habeas corpus. We certified five claims for appeal. In these claims, petitioner contends that he was denied a fair trial due to the prosecution’s wrongful suppression of Brady material; that he was denied effective assistance of counsel at trial by virtue of defense counsel’s undisclosed conflict of interest and, in the penalty phase, because counsel failed to object to hearsay evidence and failed to adequately prepare and present mitigation evidence; and that he was denied effective assistance of counsel on appeal, because counsel failed to assert claims based on trial counsel’s conflict of interest and wrongful admission of hearsay evidence at trial. For the reasons that follow, we conclude that none of the claims warrants habeas relief and we therefore affirm the judgment of the district court. I. BACKGROUND The body of Ronald Lally was found on a cemetery roadway in Cleveland on January 19,1994. Lally had suffered a gunshot wound to the head and other blunt force injuries causing death. Lally had been expected to testify for the prosecution in a criminal trial set to begin later that day. The criminal trial involved drug-trafficking charges against Raymond Smith and his son Danny Smith. After an extensive investigation, Stanley Jalowiec was indicted in March 1995 in Lorain County, Ohio, charged with murdering Lally. Raymond Smith was also charged with the murder. In a separate trial, Raymond Smith was found guilty in December 1995 and sentenced to death in January 1996. Jalowiee’s trial began in March 1996. The indictment charged Jalowiec with aggravated murder for causing Lally’s death purposely and with prior calculation and design, and included specifications for killing Lally to prevent his testimony in a criminal proceeding, and for firearm possession. The prosecution’s case was based largely on the testimony of Raymond Smith’s other son, Michael Smith, who testified that he had been present at the time of Lally’s murder, but had not participated. The defense introduced no proofs in the guilt phase of the trial. The facts shown by the trial proofs were summarized by the Ohio Supreme Court in State v. Jalowiec, 91 Ohio St.3d 220, 744 N.E.2d 163 (2001), as follows: On the morning of January 19, 1994, a partially clad male body was found in Woodland Cemetery in Cleveland. Two weeks later, the body was identified as that of Ronald Lally of Elyria. Over a year later, the grand jury indicted defendant-appellant, Stanley E. Jalowiec, for aggravated murder, with firearm and death-penalty specifications. The indictment alleged that Jalowiec purposely killed Lally to prevent him from testifying in criminal proceedings, which had been scheduled to begin on January 19, 1994. Subsequently, a jury found Jalowiec guilty as charged, and he was sentenced to death. In June 1993, Ron Lally contacted the Elyria police to volunteer as a police informant. Lally signed an agreement to become a confidential informant for the Elyria police and agreed to make controlled drug buys. On June 7, 1993, with the assistance of Officer Scott Ashley and Detective Alan Leiby, Lally made a controlled drug buy of crack cocaine from Danny Smith and his father Raymond Smith while wired with a hidden monitoring device. As a result of the controlled buy, police arrested both Raymond Smith and Danny Smith in August 1993 and charged them with aggravated drug trafficking. Both cases were eventually set for trial on January 19,1994. On January 18, 1994, the evening before the murder, Brian Howington and Jalowiec went to several bars in downtown Elyria. (Howington knew Jalowiec because Jalowiec used to visit Howington’s aunt, Joann Corrine Fike, when Howington lived with her.) Jalowiec then asked Howington to accompany him to a friend’s house on Middle Avenue. There, Howington met Ron Lally and his roommate, and the four of them smoked crack cocaine. Around 11:30 p.m., Jalowiec, Howington, and Lally went to Fike’s house and “[s]hot pool, partied some more.” About an hour later, Jalowiec got a page and asked Howington if he could borrow Fike’s car, a Chrysler LeBaron convertible. Though Howington was hesitant, he relented after Jalowiec persisted. Around 1:00 a.m., Jalowiec and Lally left Fike’s house in the LeBaron. The next time Howington saw the car was around 5:00 a.m. when Jalowiec and Raymond Smith returned it to Fike’s apartment. At that time, the car was covered with ice, and Jalowiec and Smith told Howington that the car had been washed. Fike testified that Jalowiec told her that he had washed the car because there was blood on it as a result of a fight he had had with someone at Mom’s Open Kitchen. Sharon Hopkins testified that she was at Razzle’s bar in Elyria one night in January 1994 with her brother, Terry Hopkins, Raymond Smith, Danny Smith, Michael Smith (another son of Raymond), and several others, including Jalowiec. The group stayed at Razzle’s until it closed and then, without Jalowiec, went to eat at Mom’s Open Kitchen until around 2:45-3:00 a.m. After leaving Mom’s, Sharon Hopkins rode in Danny Smith’s car with several people including Raymond, Danny, and Michael Smith. They traveled on Middle Avenue past the railroad tracks just outside the Elyria city limits and dropped Raymond and Michael Smith off by a wooded area. They drove back over the tracks and pulled into a parking lot. Approximately five to ten minutes later, a convertible drove over the tracks to where they had dropped off Raymond and Michael Smith. Danny Smith said, “That is it.” Several minutes later, the convertible drove by again heading toward town, and Danny Smith’s car began to follow it. Shortly thereafter, Danny Smith signaled the convertible to pull over and ducked down in the front passenger seat while telling Sharon Hopkins to get out and ask the people in the convertible whether they had picked up Michael Smith. Sharon Hopkins saw Jalowiec get out of the driver’s seat of the convertible. Jalowiec responded that Michael Smith was in the car. Although Sharon Hopkins could not see the other occupants, she could tell that there were four people inside the convertible. Danny Smith then drove Sharon Hopkins home. Later that morning, at around 3:3(M:00 a.m., Danny Smith arrived back at his apartment. Terry Hopkins arrived a little later and noticed that Danny Smith was “nervous and said he was feeling sick to his stomach.” Danny Smith told Hopkins that “they had done it, they did it.” Hopkins then went back to his sister’s apartment across the street from Danny Smith’s apartment. Later, Hopkins visited Danny Smith again and also saw Jalowiec, Raymond Smith, and Michael Smith. Jalowiec said, “They stomped him and ran him over with a car.” The others there indicated that “they shot him and cut him.” According to Hopkins, they were “[k]ind of like bragging about it.” Danny Smith told Hopkins they wanted this person killed because he had worn “a wire on him on a drug sale.” At approximately 9:55 a.m. on January 19, 1994, Cleveland homicide detective Michael Beaman was summoned to Woodland Cemetery. A male body had been found on a cemetery roadway. Some of the victim’s clothing was nearby in a snow bank. There was no identification on or near the victim and police did not learn the identity of the victim, Lally, until a few weeks later. Dr. Heather Raaf, a forensic pathologist with the Cuyahoga County Coroner’s Office, performed the autopsy on Lally. Dr. Raaf testified that teeth in Lally’s mouth had been knocked out by a gunshot. Dr. Raaf estimated that Lally had sustained at least eleven blows to his head and that his injuries were consistent with being stomped or struck by a vehicle several times. Dr. Raaf determined that Lally’s death resulted from a gunshot wound to the head and multiple blunt impacts to the head. The drug trafficking cases against Danny and Raymond Smith were subsequently dismissed because Lally, the primary witness in both cases, was dead. After an extensive police investigation, the grand jury indicted Jalowiec on March 8, 1995, for aggravated murder with a firearms specification. In addition, a death-penalty specification alleged that Jalowiec purposely killed Lally in order to prevent his testimony as a witness in a criminal proceeding. At trial, the key witness for the prosecution was Michael Smith, son of Raymond Smith and brother of Danny Smith. Michael Smith contacted Detective Leiby in April 1994 because he was bothered about having witnessed the Lally murder. During Raymond Smith’s murder trial, Michael Smith had been unavailable to testify, and the prosecution proffered testimony from him that had been elicited in a deposition. State v. Smith (2000), 87 Ohio St.3d 424, 428, 721 N.E.2d 93, 102. However, at Jalowiec’s trial, Michael Smith testified as a prosecution witness. Michael Smith testified that, purely by chance, he had met his father and brother at Mom’s Open Kitchen around 2:30 a.m. on the night of the murder. Raymond Smith had made a phone call and indicated to Michael Smith that he was going to leave. Michael agreed to go with his father and left with him and his brother, Danny Smith. The Smiths and Danny Smith’s girlfriend got in Danny Smith’s car and both Raymond and Michael Smith were dropped off on Middle Avenue. Raymond and Michael waited outside in the cold, even though Michael had no idea what they were waiting for. The LeBaron driven by Jalowiec with Lally as a passenger pulled up to them and stopped. Raymond Smith told Lally to get in the back seat, and Michael Smith got in the back seat on the driver’s side. Raymond Smith sat in the passenger side front seat and made introductions. Shortly thereafter, Raymond Smith brandished a gun and told Lally, “Don’t make any sudden moves.” The group stopped to buy gas, beer, and cigarettes, then drove on Route 2 toward Cleveland. Raymond Smith asked Lally, “Why did you set my son up?” Lally denied doing so, but appeared to be scared. Smith then told Lally, “We are going to give you some money, get you a bus ticket, you are going to get out of town.” During the trip into Cleveland, all four men were smoking crack cocaine. Lally agreed to leave town, and they drove to East Cleveland to buy some crack for Lally’s trip. However, they saw police cars and fire trucks in the neighborhood and decided to drive back towards downtown Cleveland. As they drove, Raymond Smith directed Jalowiec to pull the LeBaron into a Cleveland cemetery. Inside the cemetery, Raymond Smith got out of the car, put the gun to Lally’s face, and ordered him out of the car. He then told Lally, “You will never snitch on nobody again.” Michael Smith heard a gunshot and then heard Lally exclaim: “You shot me in my head, you shot me in my head.” Raymond then told Michael and Jalowiec to get out and assist him. Jalowiec got out of the car, but Michael remained inside the car and did not look out. He heard “thumps like hitting” and heard Lally plead, “I won’t tell nobody, please don’t kill me, please don’t kill me.” Michael Smith testified that after about two to five minutes of quiet, he could tell that the trunk had been opened and that his father and Jalowiec were trying to put something in the trunk. He heard someone say, “He ain’t going to fit, * * * he is too stiff,” then he heard something drop. Then Raymond Smith and Jalowiec got back in the car, and Jalowiec started the car and put it in reverse. According to Michael Smith, when Lally’s body stopped the car from going any further, Jalowiec drove forward a short distance and then put the car into reverse. Michael Smith could feel the car hit something. Jalowiec did this three times and then drove out of the cemetery. As they drove from the cemetery, Raymond Smith began arguing with Michael Smith: “This is for your brother, why didn’t you get out and help?” While driving back to Elyria, Raymond took his gun apart and threw it out the window, piece by piece. Upon arriving in Elyria, they dropped Michael Smith off at Danny Smith’s apartment. Linda Luke, a forensic serologist in the coroner’s office, conducted tests on stains found on the trunk liner of the Chrysler LeBaron. Luke testified that the DNA in Lally’s blood sample was consistent with the blood found on the trunk liner. After deliberation, the jury found Jalowiec guilty as charged. At the mitigation hearing, Jalowiec made an unsworn statement. Other witnesses also testified on Jalowiec’s behalf, including his former live-in girlfriend and several family members, including both of his parents. The prosecution presented seven witnesses in rebuttal. The jury recommended death, and the court sentenced Jalowiec to death. Id. at 168-70 (footnote omitted). Jalowiec’s various and persistent efforts to obtain appellate relief and post-conviction relief from the Ohio courts proved unsuccessful. He filed his petition for writ of habeas corpus in the Northern District of Ohio on July 2, 2003. In his amended petition, filed on November 21, 2005, Jalowiec asserted forty-seven claims for relief. In a lengthy and comprehensive opinion issued January 31, 2008, the district court denied all forty-seven claims. Jalowiec v. Bradshaw, No. l:03-CV-645, 2008 WL 312655 (N.D.Ohio). Jalowiec timely filed notice of appeal. We granted a certificate of appealability as to the following five claims: I. (Claim 20) Whether the prosecution violated Jalowiec’s rights under Brady v. Maryland, 373 U.S. 83, 83 S.Ct. 1194,10 L.Ed.2d 215 (1963), by withholding prior statements made to police by state witnesses and information regarding plea bargains and other inducements for testimony by such witnesses; II. (Claim 31) Whether Jalowiec was deprived of the effective assistance of counsel due to an alleged conflict of interest, by which counsel simultaneously represented a state witness for whom counsel obtained a sentencing benefit in return for the witness’s testimony at Jalowiec’s trial; III. (Claim 35(d)) Whether trial counsel performed ineffectively during the penalty phase by failing to object to the admission of a hearsay statement of a co-defendant; IV. (Claim 36) Whether trial counsel performed ineffectively by failing to adequately investigate and prepare mitigation evidence for the penalty phase; and V. (Claim 42(a) and (c)) Whether appellate counsel performed ineffectively on direct appeal by failing to raise the claims that Jalowiec’s trial counsel had a conflict of interest and that hearsay evidence was improperly admitted at trial. The district court determined that federal habeas review of three of these claims— claims 20, 31 and 36 — was barred by procedural default. Nonetheless, the court addressed and rejected all five claims on their merits as well. II. STANDARD OF REVIEW A. AEDPA Review We review the district court’s legal conclusions and rulings on mixed questions of law and fact de novo, and review factual findings for clear error. Boykin v. Webb, 541 F.3d 638, 642 (6th Cir.2008). Under the Antiterrorism and Effective Death Penalty Act (“AEDPA”), the federal courts may not grant habeas relief on any claim that was adjudicated on the merits in the state courts unless the adjudication resulted in a decision that: (1) was contrary to, or involved an unreasonable application of, clearly established federal law as determined by the Supreme Court; or (2) was based on an unreasonable determination of the facts in light of the evidence presented to the state courts. 28 U.S.C. § 2254(d). Under the “contrary to” clause, a federal habeas court may grant the writ only if the state court arrived at a conclusion opposite to that reached by the Supreme Court on a question of law, or if the state court decided the case differently than the Supreme Court has on a set of materially indistinguishable facts. Williams v. Taylor, 529 U.S. 362, 412-13, 120 S.Ct. 1495, 146 L.Ed.2d 389 (2000). Under the “unreasonable application” clause, a federal court may grant the writ only if the state court identified the correct governing legal principle from the Supreme Court’s decisions but unreasonably applied that principle to the facts of the petitioner’s case. Id. “[A] federal habeas court may not issue the writ simply because that court concludes in its independent judgment that the relevant state-court decision applied clearly established federal law erroneously or incorrectly.” Id. at 411, 120 S.Ct. 1495. Rather, to warrant habeas relief, the application must be found to be “objectively unreasonable.” Id. at 409, 120 S.Ct. 1495. “AEDPA thus imposes a ‘highly deferential standard for evaluating state-court rulings.’ ” Renico v. Lett, — U.S.-, 130 S.Ct. 1855, 1862, 176 L.Ed.2d 678 (2010) (quoting Lindh v. Murphy, 521 U.S. 320, 333 n. 7, 117 S.Ct. 2059,138 L.Ed.2d 481 (1997)). In analyzing whether a state court decision is contrary to or an unreasonable application of clearly established Supreme Court precedent, a federal court may look only to the holdings of the Supreme Court’s decisions as of the time that the petitioner’s state-court conviction became final. Williams, 529 U.S. at 390, 120 S.Ct. 1495; Miller v. Stovall, 608 F.3d 913, 919 (6th Cir .2010). However, the court may look to lower courts of appeals’ decisions to the extent they illuminate the analysis of Supreme Court holdings in determining whether a legal principle had been clearly established by the Supreme Court. Landrum v. Mitchell, 625 F.3d 905, 914 (6th Cir.2010). Finally, where factual findings are challenged, the habeas petitioner has the burden of rebutting, by clear and convincing evidence, the presumption that the state court’s factual findings are correct. See 28 U.S.C. § 2254(e)(1); Landrum, 625 F.3d at 914. AEDPA’s deferential standard of review applies only to state-court adjudications on the merits. Cone v. Bell, 556 U.S. 449, 129 S.Ct. 1769, 1784, 173 L.Ed.2d 701 (2009). Where AEDPA deference does not apply, state-court adjudications of legal issues are reviewed de novo and state-court fact findings are reviewed only for clear error. Id.; Evans v. Hudson, 575 F.3d 560, 564 (6th Cir.2009). Here, the district court concluded that many of Jalowiec’s habeas claims, including some of those before us now, were procedurally defaulted — that is, denied on the basis of a state procedural rule. We review the district court’s procedural default rulings de novo. Cuijetinovic v. Eberlin, 617 F.3d 833, 836 (6th Cir. 2010). B. Procedural Default Where the state courts have declined to reach the merits of a state prisoner’s federal claims due to application of “an independent and adequate state procedural rule, federal habeas review of the claims is barred unless the prisoner can demonstrate cause for the default and actual prejudice as a result of the alleged violation of federal law, or demonstrate that failure to consider the claims will result in a fundamental miscarriage of justice.” Coleman v. Thompson, 501 U.S. 722, 750, 111 S.Ct. 2546, 115 L.Ed.2d 640 (1991). A habeas petitioner’s claim will be deemed procedurally defaulted if each of the following four factors is met: (1) the petitioner failed to comply with a state procedural rule; (2) the state courts enforced the rule; (3) the state procedural rule is an adequate and independent state ground for denying review of a federal constitutional claim; and (4) the petitioner has not shown cause and prejudice excusing the default. Guilmette v. Howes, 624 F.3d 286, 290 (6th Cir.2010). In determining whether a state procedural rule was applied to bar a claim, a reviewing court looks to the last reasoned state-court decision disposing of the claim. See Ylst v. Nunnemaker, 501 U.S. 797, 803, 111 S.Ct. 2590, 115 L.Ed.2d 706 (1991); Guilmette, 624 F.3d at 291. Even if the first three factors are satisfied, the petitioner may still avoid the procedural bar and obtain merits review of his habeas claim if he demonstrates there was “cause” for his noncompliance with the procedural rule and he was actually prejudiced by the alleged constitutional error. Haliym v. Mitchell, 492 F.3d 680, 690 (6th Cir.2007). The “cause” standard requires the petitioner to show that “some objective factor external to the defense impeded counsel’s efforts to comply with the State’s procedural rule.” Id. at 690-91 (quoting Murray v. Carrier, 477 U.S. 478, 488, 106 S.Ct. 2639, 91 L.Ed.2d 397 (1986)). This may be satisfied, for instance, by a showing of interference by officials that makes compliance with the procedural rule impracticable, or attorney error rising to the level of ineffective assistance of counsel, or a factual or legal basis for a claim that was not reasonably available. McCleskey v. Zant, 499 U.S. 467, 493-94, 111 S.Ct. 1454, 113 L.Ed.2d 517 (1991). III. ANALYSIS A. Nondisclosure of Brady Materials Jalowiec contends the prosecution violated his rights under Brady v. Maryland, 373 U.S. 83, 83 S.Ct. 1194, 10 L.Ed.2d 215 (1963), by failing to disclose evidence potentially favorable to him prior to trial. He contends the prosecution wrongly withheld evidence of prior inconsistent statements made to the police by various state witnesses as well as plea-agreement deals, grants of immunity, and other inducements given to such witnesses. In Beuke v. Houk, 537 F.3d 618, 633 (6th Cir.2008), we summarized the requirements of Brady as follows: Brady requires the prosecution to disclose exculpatory and impeachment evidence that is “material either to guilt or to punishment.” Strickler v. Greene, 527 U.S. 263, 280, 119 S.Ct. 1936, 144 L.Ed.2d 286 (1999) (quoting Brady, 373 U.S. at 87, 83 S.Ct. 1194). “The evidence is material only if there is a reasonable probability that, had the evidence been disclosed to the defense, the result of the proceeding would have been different.” United States v. Bagley, 473 U.S. 667, 682, 105 S.Ct. 3375, 87 L.Ed.2d 481 (1985). A Brady violation has three elements: (1) the evidence “must be favorable to the accused, either because it is exculpatory, or because it is impeaching”; (2) the “evidence must have been suppressed by the state, either willfully or inadvertently”; and (3) “prejudice must have ensued.” Strickler, 527 U.S. at 281-82, 119 S.Ct. 1936. The district court determined that Jalowiec had failed to properly present his Brady claim (Claim 20) to the state courts and concluded that the claim was procedurally defaulted and subject to dismissal. Jalowiec, 2008 WL 312655, at *32-33. Yet, “in an effort to promote judicial efficiency and preserve judicial resources,” id. at *50, the court nonetheless addressed the merits of the claim, concluding ultimately that the prosecution ought to have disclosed some materials, but that its failure to do so was harmless error. Id. at *66-73. 1. Procedural Default of Brady Claim The procedural history of Jalowiec’s post-conviction relief efforts is convoluted and confused. To the extent Jalowiec’s Brady claim is based on the prosecution’s failure to disclose the agreement pursuant to which Michael Smith testified, the district court determined that the claim was presented to the state trial court in Jalowiec’s Third Amended Petition for Post-conviction Relief (seventh claim for relief). Id. at *32-33. Observing that the Ohio Court of Appeals affirmed the dismissal of this claim as an improper successive petition (because the third amended petition succeeded an original petition and two subsequent amendments), the district court held that this ruling represented enforcement of an independent and adequate state procedural bar that foreclosed federal habeas review of the merits. Yet, inasmuch as Jalowiec’s motion to strike his original petition and first two amendments was never addressed by the state courts — a point apparently overlooked by the Ohio Court of Appeals — there appears to be no good reason to treat the third amended petition as a successive petition, rather than as a permissible amendment of his original petition. The Ohio courts’ enforcement of the procedural bar was therefore, to this extent, erroneous and represents no obstacle to our review of this portion of Jalowiec’s Brady claim. See Cone v. Bell, 556 U.S. 449, 129 S.Ct. 1769, 1780-81, 173 L.Ed.2d 701 (2009) (holding that state court application of procedural bar resting on false premise creates no bar to habeas review of merits); Richey v. Bradshaw, 498 F.3d 344, 359-60 (6th Cir.2007) (declining to find procedural default where procedural bar was improperly enforced by state courts); Durr v. Mitchell, 487 F.3d 423, 434-35 (6th Cir.2007) (holding that incorrect application of state procedural bar was not reliance on “adequate and independent” state rule). We independently review this issue on the merits below. See Part III.A.2(f). To the extent the Brady claim is based on the prosecution’s failure to disclose prior statements of Raymond Smith, Danny Smith, Terry Hopkins, Sandra Williams, Lynne Altpater and Tammy Green, the district court held that the claim was never presented to the state courts and is therefore unexhausted and unreviewable in habeas. See 28 U.S.C. § 2254(b) (making exhaustion of state-court remedies prerequisite to federal habeas relief). Jalowiec contends this ruling is in error. First, he contends that although he did not specifically assert a Brady claim concerning the prosecution’s nondisclosure of prior statements by these witnesses in his third amended petition, the petition did include, in his fourteenth claim for relief, allegations of fact implicating undisclosed statements by Raymond Smith, Michael Smith, Sandra Williams and Lynne Altpater. Jalowiec thus impliedly argues that he adequately presented the “substance” of his Brady claim to the state courts to satisfy the exhaustion requirement. See Picard v. Connor, 404 U.S. 270, 275-78, 92 S.Ct. 509, 30 L.Ed.2d 438 (1971); Satterlee v. Wolfenbarger, 453 F.3d 362, 365 (6th Cir.2006). Yet, because of the important federal-state comity concerns served by the exhaustion doctrine, a state prisoner is required to present the state courts with the same claim, or a claim “substantially equivalent” to the claim, urged upon the federal courts. Picard, 404 U.S. at 275-78, 92 S.Ct. 509. Otherwise, the state courts are not given the opportunity to apply the controlling legal principles to the facts bearing on the constitutional claim. Id. Jalowiec’s fourteenth claim for relief asserted two different theories of relief, i.e., that the prosecution violated his due process right to a fair trial by suborning perjury, and that his trial counsel’s failure to conduct a more thorough pre-trial investigation amounted to ineffective assistance of counsel. Both of these theories of relief implicated prior statements made by prosecution witnesses, but neither gave the state courts the opportunity to apply the legal principles governing Jalowiec’s present Brady claim. It is not enough that these different claims implicated some of the same facts that are integral to Jalowiec’s Brady claim. See Anderson v. Harless, 459 U.S. 4, 6, 103 S.Ct. 276, 74 L.Ed.2d 3 (1982). It is not enough that the claims actually presented to the state courts were “somewhat similar” to the Brady claim in some respects. Id. Nor is it enough to say that the Brady “ramifications” of Jalowiec’s arguments were “self-evident.” Id. at 7, 103 S.Ct. 276. The bottom line is that the state courts were not called upon to apply the legal principles governing the constitutional claim now presented to the federal courts. Accordingly, we find no error in the district court’s conclusion that Jalowiec’s Brady claim concerning the prosecution’s nondisclosure of prior statements by these witnesses was not, by virtue of the claims asserted in the fourteenth claim for relief in his third amended petition, properly exhausted. It follows that because any attempt to now exhaust the Brady claim in the state courts would be dismissed as untimely, the claim is, to this extent, procedurally defaulted. See O’Sullivan v. Boerckel, 526 U.S. 838, 848, 119 S.Ct. 1728, 144 L.Ed.2d 1 (1999) (holding that failure to timely exhaust habeas claim amounts to procedural default). Jalowiec challenges the district court’s laek-of-exhaustion ruling on a second basis. To the extent his Brady claim is based on suppressed evidence that remained undisclosed until after the habeas proceedings were commenced, he contends the exhaustion requirement should be excused. Indeed, inasmuch as Jalowiec’s failure to timely assert certain bases for his Brady claim is shown to be attributable to the prosecution’s wrongful withholding of information, he has “cause” — i.e., some objective factor external to the defense— excusing his procedural default. Strickler, 527 U.S. at 283-89, 119 S.Ct. 1936; Beuke, 537 F.3d at 634. The Warden does not dispute that materials relevant to Jalowiec’s Brady claim were disclosed only during discovery in the district court habeas proceedings. Jalowiec has therefore adequately shown “cause.” The challenge is to determine which items are relevant to the Brady violations asserted in Claim 20 and whether their non-disclosure is shown to have resulted in such “prejudice” as to excuse the procedural default. To satisfy the prejudice requirement, Jalowiec must show there is a reasonable probability that the outcome of the trial would have been different had the materials been timely disclosed to the defense. Strickler, 527 U.S. at 289, 119 S.Ct. 1936. He must show that the withholding of materials “worked to his actual and substantial disadvantage, infecting his entire trial with errors of constitutional dimensions.” Beuke, 537 F.3d at 634 (quoting Jamison v. Collins, 291 F.3d 380, 388 (6th Cir.2002)). This prejudice inquiry mirrors the Brady materiality analysis implicated by the merits of the claim. Id. We therefore look to the district court’s assessment of the merits of Claim 20 as our starting point in determining whether Jalowiec has shown prejudice excusing his procedural default of his claim to the extent it is based on nondisclosure of witness statements. We also look to the district court’s assessment of the merits in regard to Jalowiec’s claim that the prosecution failed to disclose an agreement pursuant to which Michael Smith testified because that aspect of his claim was not procedurally defaulted. 2. Prejudice: Materiality of Withheld Information Because the state courts did not address the merits of the Brady claim, we are not constrained by AEDPA deference; like the district court below, we undertake de novo review of materiality and prejudice. See Cone v. Bell, 556 U.S. 449, 129 S.Ct. 1769, 1784, 173 L.Ed.2d 701 (2009). Again, in order to make out a Brady violation, Jalowiec must show that (1) evidence favorable to the defense, (2) was suppressed by the government, and (3) the defense was prejudiced. In determining whether “withheld information was material and therefore prejudicial,” a reviewing court considers “it in light of the evidence available for trial that supports the petitioner’s conviction.” Jells v. Mitchell, 538 F.3d 478, 502 (6th Cir.2008). “[E]vidence is ‘material’ within the meaning of Brady when there is a reasonable probability that, had the evidence been disclosed, the result of the proceeding would have been different.” Cone, 129 S.Ct. at 1783. “[A] showing of materiality does not require demonstration by a preponderance that disclosure of the suppressed evidence would have resulted ultimately in the defendant’s acquittal.” Kyles v. Whitley, 514 U.S. 419, 434, 115 S.Ct. 1555, 131 L.Ed.2d 490 (1995). Favorable evidence is material if it “could reasonably be taken to put the whole case in such a different light as to undermine confidence in the verdict.” Cone, 129 S.Ct. at 1783 (quoting Kyles, 514 U.S. at 435, 115 S.Ct. 1555). If materiality of the suppressed evidence is established, making out a Brady violation, harmless-error inquiry does not apply. Kyles, 514 U.S. at 435, 115 S.Ct. 1555. In reviewing for materiality, we consider the effect of the suppressed evidence “collectively,” rather than “item by item.” Cone, 129 S.Ct. at 1785 (quoting Kyles, 514 U.S. at 436, 115 S.Ct. 1555). Nonetheless, we begin, like the district court, by considering each item asserted in Claim 20 individually- (a) Danny Smith Danny Smith, one of the defendants (along with Raymond Smith) against whom Ronald Lally had been set to testify in connection with drug-trafficking charges just before he was killed, gave a recorded statement to police during custodial interrogation on January 7, 1995. Danny Smith told Elyria Police Detective Alan Leiby, investigating the Lally murder, that he “never really suspected of [sic] Stan [Jalowiec] would do something like that until everything started coming in and people start [sic] bringing up his name.” This statement was not disclosed to Jalowiec prior to or at the time of his trial. Although Danny Smith, as a suspected co-conspirator, was “unavailable” to testify at Jalowiec’s trial, Jalowiec argues the statement should have been disclosed as Brady material because it tended to exonerate him, refuting the prosecution’s theory that he was involved in a suspected conspiracy to kill Lally. The district court summarily held that Danny Smith’s January 7, 1995 statement was “material” because, had it “been disclosed prior to trial or at trial, the favorable evidence could reasonably be taken to put the whole case in such a different light as to undermine confidence in the verdict.” Jalowiec, 2008 WL 312655, at *67. However, the court ultimately concluded that this “Brady violation,” viewed in light of all the evidence implicating Jalowiec, was harmless beyond a reasonable doubt. Id. at *73. On appeal, Jalowiec correctly contends the district court’s reasoning is self-contradictory. He contends the district court, having once found that the statement was “material,” and its nondisclosure a “Brady violation,” was not at liberty to engage in harmless-error review to deny habeas relief. Indeed, the district court expressly recognized earlier in its opinion that “once a Brady violation is found, there is no need for further harmless-error review[;] ... [a] Brady violation is never harmless.” Id. at *67 (citations omitted). The Warden does not deny the facial inconsistency. However, in view of the court’s unambiguous denial of habeas relief, the Warden contends the opinion should be construed as concluding that Danny Smith’s statement was potentially exculpatory and should have been disclosed per Brady, but was not, ultimately, “material.” The Warden’s position finds support not only in a common-sense reading of the opinion as a whole, but also in the Supreme Court’s recognition that “not every violation of [the prosecution’s duty of disclosure] necessarily establishes that the outcome was unjust.” Strickler, 527 U.S. at 281, 119 S.Ct. 1936. As the Strickler Court further observed: [T]he term “Brady violation” is sometimes used to refer to any breach of the broad obligation to disclose exculpatory evidence — that is, to any suppression of so-called “Brady material” — although, strictly speaking, there is never a real “Brady violation” unless the nondisclosure was so serious that there is a reasonable probability that the suppressed evidence would have produced a different verdict. Id. Consistent with this observation, we read the district court’s opinion, despite its misleading language, as concluding that the prosecution breached its duty of disclosure under Brady, but that the Danny Smith statement was not so “material” that its timely disclosure to the defense would have given rise to a reasonable probability of a different verdict. This is a conclusion that we also reach de novo. We find no error in the district court’s determination that the January 7, 1995 statement should have been disclosed. Yet, the statement is arguably as incriminating as it is exculpatory. Jalowiec has not demonstrated — even assuming he could have found a way of introducing Danny Smith’s hearsay opinion about a friend’s character — that he could have made such effective use of it as would put the whole case in such a different light as to undermine confidence in the verdict. Nor has he explained how the statement might have led to other admissible exculpatory evidence. The statement is therefore not “material” for Brady purposes, and its nondisclosure was not so prejudicial as to amount to a “Brady violation.” In addition to the January 7, 1995 statement, the prosecution withheld recorded conversations between Danny Smith and prosecution witness Carl Hartman in which Danny attempted to bribe Hartman and influence him as a witness. Jalowiec contends that Danny’s statements could have been used to cross-examine Hartman and to undermine the prosecution’s theory that Jalowiec conspired to kill Lally by tending to show Danny Smith’s greater involvement and culpability. The district court held this evidence was neither exculpatory nor material. Jalowiec, 2008 WL 312655 at *68. On appeal, Jalowiec has failed to demonstrate that the district court erred in either respect. Jalowiec has not demonstrated hoiv he could have effectively used such evidence of Danny Smith’s efforts to influence Hartman to undermine confidence in the jury’s assessment of the evidence against him. (b) Terry Hopkins Terry Hopkins was staying in an apartment with Danny Smith at the time of the Lally murder. He was out drinking with Danny Smith and others in the early morning hours on the day of the murder, January 19, 1994. He gave inconsistent recorded statements to officers on January 4 and 6, 1995, concerning conversations he had with Danny Smith, Raymond Smith and Jalowiec in the hours after the murder. Although Hopkins first stated that Jalowiec admitted shooting Lally, he later stated that Jalowiec identified Raymond Smith as the shooter. The district court concluded that these statements should have been provided to Jalowiec, but that they were not material because Hopkins also made various inconsistent statements at trial and was subject to effective cross-examination. Jalomec, 2008 WL 312655, at *68. We find no error in the district court’s assessment. On cross-examination, Hopkins readily admitted that he gave multiple statements to the police. He acknowledged that he was not very cooperative on one occasion because he “didn’t want to get involved.” He did not deny that he subsequently gave a “revised” statement. Hopkins seemed unable to remember when he gave specific statements to the police. He admitted he had drunk heavily the night Lally was killed and was “pretty drunk” when he heard the conversations he reported on. Thus, it is apparent that even without the benefit of specific inconsistent statements from undisclosed police interviews, Jalowiec’s counsel managed to effectively cross-examine Hopkins, highlighting weaknesses in his testimony. Although we agree with the district court’s determination that Hopkins’ prior inconsistent statements should have been disclosed, Jalowiec has not demonstrated that his efforts to impeach Hopkins would have been so significantly aided by the statements as to create a reasonable probability that the outcome of the trial would have been different. The nondisclosure of these statements was therefore not material under Brady. (e) Lynne Altpater Lynne Altpater was an acquaintance of Jalowiec and Danny Smith who testified for the prosecution in Jalowiec’s trial. She testified that she was working at a doctor’s office in December 1993 when Danny Smith approached her and asked if she could get him some poison to “take care of a person” who could cause him to be in a lot of trouble. She had previously given recorded statements to the police on January 6, 1995 and January 19, 1995 — statements that were not disclosed to the defense. Her testimony was consistent with her prior statement of January 19, but inconsistent with her January 6 statement, when she told Detective Leiby that Danny Smith wanted the poison to kill some cats that had crawled on his car at work. Although Jalowiec could have used the January 6 version as a prior inconsistent statement to impeach Altpater, the district court concluded that the statement was not material for Brady purposes. The court reasoned that Altpater’s testimony was not that damaging to Jalowiec, considering (a) there was no evidence that he was privy to the conversation between Altpater and Danny, and (b) “poison was not used to kill Lally.” Jalowiec, 2008 WL 312655, at *69. We find no error in this assessment. Altpater’s testimony was useful to the prosecution to show that Danny Smith, one of the persons Jalowiec was suspected of conspiring with, had explored at least one other way of preventing Lally from testifying against him and his father on the then pending drug-trafficking charges. Otherwise, it had no tendency to incriminate Jalowiec. Further, even if Jalowiec could have used the January 6 statement to attack Altpater’s credibility, the prosecution could have used the January 19 statement to rehabilitate her by showing that Altpater had gradually become more cooperative and forthcoming. Considering the broader context, the January 6, 1995 statement was not so material as to warrant a finding that its disclosure to the defense would have made a difference. Jalowiec asserts a second Brady violation in relation to witness Altpater. At Danny Smith’s trial, subsequent to Jalowiec’s, prosecution witness Lynne Altpater testified that Detective Leiby had offered her husband, Richard Altpater, assistance with drug charges pending against him in exchange for his cooperation with the prosecution of Danny Smith. That is, according to a statement recorded by attorney Michael Duff on October 23, 1995, Richard Altpater said Detective Leiby asked him to corroborate information obtained from other witnesses to the effect that Danny Smith had offered him, Richard Altpater, money to kill a person or take him out of town. Richard Altpater purportedly told Leiby he would cooperate, even though, in truth, Danny Smith had not offered him money, but had only asked if he knew anyone who could hurt someone for money. In his recorded statement, Richard Altpater went on to explain that he “used to whoop people’s ass for money,” but that he’s “getting too old for that.” As it turned out, Richard Altpater was not called to testify and Leiby did not provide assistance with Richard Altpater’s pending charges. Nonetheless, Jalowiec argues that the prosecution should have disclosed evidence of Leiby’s offer to induce Richard’s cooperation for his use in impeaching Lynne Altpater’s testimony in his trial. The district court held that information relating to Leiby’s offer should have been disclosed to Jalowiec, but that the nondisclosure did not result in cognizable prejudice. Jalowiec, 2008 WL 312655, at *72-73. Although the district court carelessly referred to the nondisclosure of the information as a “Brady violation,” Jalowiec has not otherwise demonstrated error -by the district court. Jalowiee has failed to identify evidence that Lynne Altpater’s testimony in his case was actually induced or influenced by any promise or offer of assistance with her husband’s charges. Even though she actively participated in the statement recorded on October 23, 1995, the statement includes no suggestion that she was influenced by Leiby’s offer to her husband. Moreover, as indicated above, Lynne Altpater’s testimony in Jalowiec’s trial about Danny Smith’s request for poison — irrespective of whether he couched his request in terms of a need to kill a cat or a person — was of marginal relevance in establishing Jalowiec’s guilt. There is no reasonable basis to conclude that the disclosure of Leiby’s offer would have affected the verdict. (d) Joann Fike Joann Fike was the owner of the Chrysler LeBaron that Jalowiee borrowed on the night of Lally’s murder. She testified for the prosecution that her nephew Brian Howington let Jalowiee borrow the car, and that it was covered with ice and had blood on it when Jalowiee and Raymond Smith returned it several hours later. After the LeBaron was impounded in connection with the murder investigation, Detective Leiby helped Joann Fike find another car, gave her $200 for a set of four tires, and helped her obtain the car’s return. Jalowiee contends Leiby’s help represents evidence that he induced Fike’s cooperation, evidence that should have been disclosed. The district court acknowledged that the help Leiby gave may or may not have been usual protocol, but concluded that this information should have been disclosed to Jalowiee. Jalowiee, 2008 WL 312655, at *73. Nonetheless, the district court concluded the error was “harmless.” Id. . Jalowiee argues that any inducement given to Fike was important because she was a key witness who placed Jalowiee in her car on the night of the homicide. The Warden concedes that the economic consideration Fike received was relevant to her credibility, but argues that the evidence would not have altered the outcome of the trial if it had been given to the defense. We agree. Fike’s testimony was corroborated by other witnesses. There is no good reason to question her credibility. Even though the evidence of help Fike received should have been disclosed, Jalowiec has presented no reason to believe the information could have been used to so impeach Fike as to call the fairness of the trial and its outcome into question. We also find no error in the district court’s determination that evidence that Fike was granted transactional immunity in exchange for her cooperation with the investigation of Lally’s death was not material for purposes of Brady. Again, the grant of immunity should have been disclosed, but inasmuch as Fike did not reveal any self-incriminating information, the impeachment value of the immunity agreement was minimal. See Marshall v. Hendricks, 307 F.3d 36, 56 (3d Cir.2002) (recognizing that “the impeachment value of the immunity agreement is inextricably tied to the self-incriminating evidence that was provided after the immunity agreement was executed.”). (e) Sharon Hopkins Jalowiee contends the prosecution withheld evidence that Sharon Hopkins’s testimony was induced by Detective Leiby’s having obtained a signature bond for her boyfriend after he turned himself in on misdemeanor warrants. The district court concluded that this information was not material because the charges were minor and the boyfriend would have “more than likely” obtained a bond without Leiby’s help. Jalowiec, 2008 WL 312655, at *72. Sharon Hopkins’s testimony was important to the prosecution. She corroborated Michael Smith’s description of the transfer of passengers (Raymond and Michael Smith) from Danny Smith’s car to Fike’s LeBaron (containing Jalowiec and Lally) in the middle of the night, a transfer that linked Howington’s and Fike’s testimony about the LeBaron and linked Jalowiec to Lally throughout the night. Although the prosecution improperly withheld the information that Leiby assisted Hopkins’s boyfriend, we affirm the district court’s assessment that the information would not have undermined confidence in the outcome if it had been made available to Jalowiec. Considering that the benefit accrued to a boyfriend and not Sharon Hopkins herself, that minor charges were involved, and that Jalowiec has not disputed that the boyfriend might well have received the bond regardless of Leiby’s intercession, Jalowiec has not shown that the information was material under Brady. (f) Michael Smith Michael Smith was the prosecution’s star witness. Jalowiec contends that the prosecution withheld: evidence that Michael called Detective Leiby in 1993 or 1994 and tried to take the blame for the drug charges then pending against Danny Smith and implied that he might “do something” when Leiby rejected his story; evidence that Michael called Leiby in February 1996 and told him that he knew nothing about a conspiracy between Danny and Raymond Smith relating to Lally; and comments in police files reflecting belief that Michael had a “violent nature.” Jalowiec also contends the prosecution withheld information suggesting a connection between Michael’s testimony against Jalowiec and lenient sentencing treatment Michael received on an unrelated offense. The district court discounted the supposedly withheld information that Michael Smith had been granted favorable treatment in exchange for his testimony at Jalowiec’s trial. See Jalowiec, 2008 WL 312655, at *72. The court noted that on November 22, 1994 — long before Jalowiec’s trial — Michael was given shock probation on a five-to-fifteen-year prison sentence for a crime unrelated to the events of Jalowiec’s case. Then, after Michael testified, he received an early discharge from probation. Id. The district court correctly found there was no evidence, only speculation, linking the two. Further, as the district court determined, id. at *69, Michael Smith’s telephonic denial of knowledge of any conspiracy between his brother and father was completely consistent with his trial testimony, in which he expressed shock and surprise at what was happening during the early morning hours of January 19, 1994. The statement thus had little if any impeachment value and was not otherwise exculpatory of Jalowiec. Similarly, the cited comments about Michael’s suspected violent nature were unsubstantiated and, even if admissible, bore little if any exculpatory or impeachment significance. They can hardly be considered material under Brady. (g) Tammy Green (Castro) Jalowiec contends that Tammy Green, Danny Smith’s girlfriend, was given transactional immunity for her grand jury testimony and that this fact was not disclosed to the defense. The district court correctly concluded that, even if Tammy Green was given immunity for her grand jury testimony, the prosecution was under no obligation to disclose it because she did not testify at Jalowiec’s trial. Jalowiec, 2008 WL 312655, at *72 (citing United States v. Mullins, 22 F.3d 1365, 1372 (6th Cir.1994) (observing that because the prosecution did not call the agreement’s beneficiary as a witness at trial, “there was no one to impeach.”)). Further, Jalowiec maintains that Green’s grand jury testimony was contrary to Sharon Hopkins’s trial testimony and that he could have used it to impeach Hopkins if it had been disclosed to him. The district court reasoned that the prosecution had no duty to disclose the grand jury testimony, despite its potential impeachment value, because there is no showing that Jalowiec could not have obtained Green’s story from her directly and called her as a witness. Id. (citing Carter v. Bell, 218 F.3d 581, 601 (6th Cir.2000) (holding there was no Brady violation where the defendant should have known of the information and it was available from another source)). See also Doan v. Carter, 548 F.3d 449, 460 (6th Cir.2008) (same). On appeal, Jalowiec argues the district court’s reasoning is flawed. He contends he had no reason to know Green’s version of the events on January 19, 1994 was different than Sharon Hopkins’s and Michael Smith’s. Yet, Green’s presence among those involved in the events of January 19 was attested to by others. Her recollection of the events should have been readily discoverable with minimal investigation by defense counsel. Under such circumstances, “the Brady rule does not assist a defendant who is aware of essential facts that would allow him to take advantage of the exculpatory evidence at issue,” such as when the evidence in question “would have been discoverable with minimal investigation by [defense] counsel.” Coleman v. Mitchell, 268 F.3d 417, 438 (6th Cir.2001); see also Benge v. Johnson, 474 F.3d 236, 243 (6th Cir .2007). Moreover, in her grand jury testimony, Green testified not only that she did not know who Sharon Hopkins was, but also that Michael Smith was not in the car with her and Danny that night. She also stated that she did not see Fike’s LeBaron that night. Her version of events before she and Danny and Raymond and Jalowiec parted ways on January 19, read in isolation, would have assisted Jalowiec’s defense insofar as it contradicted the testimony of two prosecution witnesses on some background details. Generally, however, Green made clear her unwillingness to get involved. She admitted that she wanted to remain “willfully ignorant” of Danny’s criminal activities. Hence, even if the prosecution ought to have disclosed the substance of her grand jury testimony, it is unlikely that the account of such a reluctant witness would have effectively impeached the testimony of Michael Smith and Sharon Hopkins. Accordingly, we find no error in the district court’s ruling. (h) Alternate-Suspects Evidence Jalowiec contends that the prosecution withheld “significant, credible evidence regarding alternate suspects — suspects who actually had a motive to kill Lally.” Jalowiec identifies several trial witnesses, including all three Smiths, Terry Hopkins, and Vernard Berry, who, unlike him, were “involved in the drug world.” Jalowiec also refers to an undisclosed transcript of a November 29, 1995 interview with a Melissa Arroyo, who told officers that Danny Smith had told her that he, Danny, had killed Lally. Finally, Jalowiec refers to several potential suspects named in various police reports as being close to Danny Smith or having involvement with Lally. These bases for Jalowiec’s Brady claim were not asserted in his amended habeas petition, but only in his traverse. It is apparently for this reason that the district court properly declined to address these asserted grounds for relief. See Tyler v. Mitchell, 416 F.3d 500, 504 (6th Cir.2005) (holding that district court did not err in declining to address claim first raised in traverse rather than in habeas petition). Even if we were to find these grounds properly presented, they would be unavailing. Jalowiec’s identification of trial witnesses as “alternate suspects” is based largely on their trial testimony and is not undisclosed Brady evidence at all. Further, the mere fact that these witnesses were allegedly “involved in the drug world” is insufficient to render them “suspects” in Lally’s murder. According to Melissa Arroyo’s statement, Danny Smith told her that he killed somebody in “the fall season;” that “they beat the crap out of him and then um, they tied his feet to the bumper and they drove him up and down the street, and then they ran him over.” Danny did not tell her who the person was and she had no idea whether Danny was referring to Lally. Although this statement might be deemed Brady material, Jalowiec has not shown how it could have been used, in view of the other trial evidence, to call the integrity of the verdict into question. The remaining alternate-suspects “evidence” Jalowiec cites is so vague or peripheral to the facts of the Lally murder that it can hardly be deemed exculpatory, much less material. (i) Jailhouse Informant Evidence Jalowiec contends that the prosecution revealed only after trial that Detective Leiby had arranged for the placement of “snitches,” including Danny Smith, in his pretrial cell “in a vain attempt to procure admissions.” He contends that he made no incriminating statements and that he should have been able to use this information at trial to discredit the prosecution. This theory of relief was not asserted in Jalowiec’s petition either, but only in his traverse. It was not addressed by the district court and is not properly before us. See Tyler, 416 F.3d at 504. The claim is patently meritless in any event because Jalowiec has offered no rationale whereby his asserted silence or failure to make an incriminating statement could be deemed exculpatory “evidence” that the prosecution was obliged to disclose under Brady. (j) Sentencing Evidence In support of his claim that the prosecution violated his Brady rights at sentencing, Jalowiec cites an undisclosed police report showing that prosecution witness Jeff Hicks gave inaccurate testimony in the penalty phase. In his testimony, Hicks said that as he left a bar in Elyria on the night of January 27-28, 1994, he was struck by Jalowiec “with a bat, fist, or some type of club,” that he was knocked out, and that, according to a friend, Jalowiec kicked him in the face while he was unconscious. This account differed from the statement he made to the police two days after the assault only in that Hicks then reported that as he exited the bar, he was “either pushed, or fell to the ground, and passed out” before Jalowiec kicked him in the face, fracturing his jaw. Nonetheless, the police report shows that the focus of the ensuing investigation was on Jalowiec’s having kicked Hicks in the face while he lay unconscious and defenseless. The district court concluded that the police report could “not reasonably be taken to put the whole case in such a different light as to undermine confidence in the verdict.” Jalowiec, 2008 WL 312655, at *70. This handling of the Hicks evidence was clearly appropriate. The most damning aspect of Hicks’s sentencing testimony was not so much the manner in which he was initially knocked out, but that Jalowiec kicked him in the face and fractured his jaw while he was lying defenseless on the ground. Given that Hicks’s testimony suggested that he was “blindsided,” and that the police report indicated Hicks did not know how he fell unconscious to the ground, the police report would have had little impeachment value to Jalowiec and can hardly be deemed material under Brady. (k) Materiality of Withheld Information: Conclusion Accordingly, we hold that none of the allegedly suppressed items of information is so “material” for Brady purposes as to give rise to a reasonable probability that the outcome of the trial would have been different if it had been disclosed to Jalowiec prior to trial. In assessing materiality, however, we must consider the effect of the suppressed evidence “collectively.” Cone, 129 S.Ct. at 1785. The sheer number of undisclosed, potentially exculpatory items in this case suggests a troubling disregard by the prosecution of its Brady obligation, which we do not condone. Our role on habeas review, however, is not to police the prosecution, but to ascertain whether the prosecution’s failure to disclose Brady material compromised Jalowiec’s right to a fair trial. Almost all of the undisclosed information identified by Jalowiec has potential impeachment value; it is not directly exculpatory. Exculpatory evidence is not inherently more valuable, because the Brady materiality prong is not a sufficiency-of-the-evidence test, but there are relevant distinctions between impeachment and exculpatory evidence for Brady purposes. For instance, “[w]here the undisclosed evidence merely furnishes an additional basis on which to challenge a witness whose credibility has already been shown to be questionable or who is subject to extensive attack by reason of other evidence, the undisclosed evidence may be cumulative, and hence not material.” Robinson v. Mills, 592 F.3d 730, 736 (6th Cir.2010) (internal quotation marks omitted). Defense counsel thoroughly cross-examined several of the prosecution witnesses who are the subjects of Brady impeachment evidence in this case. Weaknesses and inconsistencies in the prosecution’s case were exposed. The jury was well aware that most of the prosecution witnesses were not model citizens and many were under the influence of intoxicants at the time of the events they testified about. The undisclosed evidence J