Rodney Reed walks into the courtroom for a hearing at the Bastrop County Criminal Justice Center in Bastrop on Tuesday November 25, 2014.
CNN  — 

The Supreme Court struggled with the case of Rodney Reed on Tuesday, a Black death row inmate seeking post-conviction DNA evidence to prove his innocence. He claims an all-White jury wrongly convicted him of killing a White woman in Texas in 1998.

Since his conviction, Texas courts have rejected his various appeals. Celebrities such as Kim Kardashian and Rihanna have expressed support, signing a petition asking the state to halt his execution.

At oral arguments, a lawyer for the state suggested Reed was making his arguments in order to delay his execution, but it was unclear if the lawyer swayed all of the conservative justices. The state warned against inmates being able to “avail themselves” of “endless procedure in state courts.”

The case puts a new focus on the testing of DNA crime-scene evidence and when an inmate can make a claim to access the technology in a plea of innocence. To date, 375 people in the United States have been exonerated by DNA testing, including 21 who served time on death row, according to the Innocence Project, a group that represents Reed and other clients seeking post-conviction DNA testing to prove their innocence.

Justice Elena Kagan seemed to agree with Reed’s attorney, saying it was important for the full appeals process to play out before an inmate asked a federal court to intervene.

“Isn’t the simplest thing just to say that the person isn’t harmed until the state process has come to an end and we know for a fact what the state judgment is?” she asked.

Justice Ketanji Brown Jackson also agreed, noting that if an inmate were to bring the challenge before the state proceedings were over, a federal court would likely move to put the appeal on hold until the state action concludes.

But Chief Justice John Roberts expressed concern at one point for inmates seeking to avoid an immediate appeal as a way to “put off” an execution.

For his part, Justice Brett Kavanaugh was interested in the “practical problems” courts would face if an inmate had to bring a federal court challenge before the state procedures were exhausted.

Court precedent allows a state prisoner who has been denied in state court to pursue a post-conviction claim for DNA testing in federal court. But Reed’s case raises a statute of limitations question about whether such a claim can be brought at the end of state court litigation or at the moment a trial court denied DNA testing.

Lower courts have split on the issue and the distinction is key for Reed as a federal appeals court ruled that he waited too long to bring his claim. How the Supreme Court rules could impact other death row inmates across the country seeking to test new evidence. And comes as DNA testing has become a more utilized means to exonerate those who have been wrongly convicted.

Reed has been on death row for the murder of 19-year-old Stacey Stites.

A passerby found Stites’ body near a shirt and a torn piece of belt. Investigators targeted Reed because his sperm was found inside her. Reed acknowledged the two were having an affair, but says that her fiancé, a local police officer named Jimmy Fennell, was the last to see her alive.

Reed claims that over the last two decades he has discovered a “considerable body of evidence” demonstrating his innocence. Reed claims that the DNA testing would point to Fennell as the murder suspect. Fennell was later jailed for sexually assaulting a woman in his custody and Reed claims that numerous witnesses said he had threatened to strangle Stites with a belt if he ever caught her cheating on him. Reed seeks to test the belt found at the scene that was used to strangle Stites.

The Texas law at issue allows a convicted person to obtain post-conviction DNA testing of biological material if the court finds that certain conditions are met. Reed was denied. He came to the Supreme Court in 2018 and was denied again. Now he is challenging the constitutionality of the Texas law arguing that the denial of the DNA testing violates his due process rights.

But the 5th US Circuit Court of Appeals held that he waited too long to bring the claim. “An injury accrues when a plaintiff first becomes aware, or should have become aware, that his right had been violated.” The court said that he became aware of that in 2014 and that his current claim is “time barred.”

Reed’s lawyers argued that he could only bring the claim once the state appeals court had ruled, at the end of state court litigation. In court, Parker Rider-Longmaid said that the “clock doesn’t start ticking” until state court proceedings come to an end. He said Texas’ reading of the law would mean that other procedures in the appellate process are “irrelevant.”

Janai Nelson of the NAACP Legal Defense Fund supports Reed arguing in briefs that the “overwhelming majority of incarcerated persons exonerated through DNA evidence since its introduction in 1989, have been people of color and primarily Black men.”

Texas Solicitor General Judd Stone II responded in court papers that at trial the state introduced “substantial” evidence that Reed had “sexually assaulted multiple other women.” Besides arguing that the suit is untimely, he also says that Reed’s claim cannot be brought because he lacks the legal right, or “standing,” to bring the case.