US Supreme Court to overturn Roe vs Wade

Legally, the AZ AG is correct. There is a post conviction process in place to address such things, which is the meat of the problem SCOTUS was addressing. Did the prisoners correctly follow that process? There are not decades of precedent here which are being overturned, the problem is that there is no precedent at all. The argument is over whether the Martinez holding can be read to include the Shinn case or not. From SCOTUSblog.

In Shinn v. Ramirez and Jones, two men on Arizona’s death row raised claims in habeas corpus proceedings that their trial attorneys were constitutionally ineffective – one for failing to investigate evidence suggesting his client could not have committed the crime, and the other for failing to investigate her client’s intellectual disability, which could have spared him the death penalty. Although the Supreme Court’s 2012 decision in Martinez v. Ryan permitted defendants to raise such claims for the first time in federal court, on Monday the court ruled 6-3 that they cannot develop evidence to support those claims.

For those who want to read the actual opinions, both majority and dissent , it can be found here. https://www.supremecourt.gov/opinions/21pdf/20-1009_19m2.pdf

It’s over 40 pages long, and is not a simple matter. I certainly won’t offer an opinion as to which side is correct as I don’t have the hours to spend looking up the case law and legislation cited by both sides, but I will observe that none of the nine judges care (in their written opinions, at least) whether or not the prisoners are actually innocent. That’s not the federal judiciary’s role, they only care that federal laws are applied in the determination of guilt or innocence.

It's not their job to provide a relief valve. Guilt or innocence of a state crime is not within the jurisdiction of the federal system. Even POTUS cannot provide any relief like a pardon, the federal oversight on the matter is purely on procedural matters. Does the AZ system fall within the procedural limitations and protections provided by the Constitution and Congress? If so, the federal hands are tied. The dispute between the majority and dissent is purely on the matter of this question.

Surely that is the single most important reason for over-turning a conviction, particularly a sentence of death??

No, it's the most important reason for finding a mechanism for overturning the conviction, however. The legal reasoning would be something like "Newly discovered evidence" or "error at trial." Determination of guilt or innocence is up to a jury. Then there are other options other than post-conviction motions to stay sentences, up to governors' pardons.

Not quite. The Arizona process does allow such new argument to be brought up as part of the post-conviction mechanism. If your trial lawyer screwed up, you're supposed to bring it up when you get your post-conviction hearing.

The problem here is that of the various arguments presented by the prisoners at the post-conviction hearing, they left out a few which they then decided they wanted to bring up later in the process. This is a fairly basic principle in US jurisprudence, there's an evidentiary phase which is normally the lower court, and once the evidentiary phase is complete, it's generally too late to bring in new argument for the appeals phases which revolves only around matters of law, not of fact (Which is why sometimes cases will be remanded to lower courts for factual determinations.

From the opinion

"And the doctrine of procedural default—“an important ‘corollary’ to the exhaustion requirement,” Davila v. Davis, 582 U. S. ___, ___—generally prevents federal courts from hearing any federal claim that was not presented to the state courts “consistent with [the State’s] own procedural rules,”  "

This is different from the idea that "new evidence has been discovered" which will allow a reboot of the process, but in this case, there was no 'new evidence discovered', because the facts were known before the post-conviction hearings, just not entered into the record. In effect, you're arguing "We lost the trial because my lawyer was incompetent, and we need a review" followed by "We lost the review because my new lawyer was also incompetent, we need a review to the review." Which, in theory, can go on ad infinitum.

The bottom line, though, is that this is a procedural problem for Arizona to sort out, not the Federal Court.

It's worth noting the Federal law on the matter which lays out when the prisoner can appeal to the federal courts, https://www.law.cornell.edu/uscode/text/28/2254

God be thanked. A ruling that should never have been made based on the penumbra of un-enumerated rights has been revoked.