(CNN) -- A Texas state board is set Friday to revisit questions surrounding a controversial 2004 execution, with supporters of the man's family warning the panel is trying to bury its own critical review of the case.

Cameron Todd Willingham was executed in 2004 for a fire that killed his three daughters. Prosecutors argued that Willingham deliberately set the 1991 blaze -- but three reviews of the evidence by outside experts have found the fire should not have been ruled arson.

The last of those reports was ordered by the Texas Forensic Sciences Commission, which has been looking into Willingham's execution since 2008. But a September 2009 shake-up by Texas Gov. Rick Perry has kept that panel from reviewing the report, and the commission's new chairman has ordered a review of its operating rules. Critics say that may kill the probe.

"They are attempting permanently to keep the investigation from continuing and moving on, and I do believe it's because they don't like the direction the evidence is leading," Willingham's cousin, Pat Cox, said Thursday.

The Forensic Science Commission's chairman is now John Bradley, an Austin-area district attorney with a reputation as a staunch supporter of the death penalty. Bradley has pledged to state lawmakers that the Willingham investigation "absolutely" will continue -- but said the panel needs better rules to guide its work, and could not say when the Willingham issue would move forward.

Thursday, he told CNN that concerns of Willingham's supporters were based on "a lot of misinformation."

"I think that's being used very much as a side issue to politicize, through some New York lawyers, the work of the commission," Bradley said. "The commission has been very clear that the commission is going to address the merits of the Willingham case."

The panel meets again Friday in Houston, and one of the items on its agenda is a legal opinion arguing that the panel has "relatively narrow investigative jurisdiction." The unsigned memorandum argues that the commission's mandate covers only cases on which a state-accredited forensic laboratory worked.

But because Texas started accrediting crime labs in 2003, Cox and others who have backed the family say that would mean cases such as Willingham's and that of another inmate, Ernest Willis, would be dropped. State Sen. Rodney Ellis, who pushed for the commission's creation, calls the opinion flawed.

The Forensic Sciences Commission "was operating within the language and intent of the law when it determined that it had jurisdiction to investigate the case the first time in August 2008," Ellis said in a written statement to CNN. "Frankly, I am surprised that the commission is even questioning whether or not it has jurisdiction, since it unanimously decided -- with the attorney general's representative in the room -- to review the cases over two years ago."

Ellis, a Houston Democrat, serves as the chairman of the board of The Innocence Project -- the "New York lawyers" that have supported efforts by Willingham's stepmother and cousins to clear his name. The group advocates for prisoners it says are wrongly convicted, and Ellis said the commission's work "is too important to be bogged down in political bickering."

"Texans need the FSC to perform its work in a timely manner, so the public can once again have confidence in forensic evidence and confidence that the truly guilty are behind bars and the innocent are free," he said.

But Bradley said the commission has never decided to apply the logic of the legal opinion to the case on Friday's agenda.

Bradley was named the panel's chairman two days before the Forensic Sciences Commission was to hear from Craig Beyler, a Maryland-based fire science expert. Beyler concluded the arson finding at the heart of the Willingham case "could not be sustained," either by current standards or those in place at the time.

The Innocence Project requested the investigation after a report it commissioned reached the same conclusion. Death-penalty opponents say an impartial review of Willingham's case could lead to the unprecedented admission that the state executed an innocent man.

Perry, who signed off on Willingham's execution, is up for re-election in November, and his critics have accused him of trying to short-circuit that review. Perry has said he remains confident of the condemned man's guilt, and police in the town of Corsicana, where the fire occurred, say other evidence beyond the arson testimony Beyler criticized supports the prosecution.

Cox, a retired nurse in Ardmore, Oklahoma, told CNN that spiking the commission's investigation would be a "blatant miscarriage of justice."

"The reasonable people of this country and the state of Texas can see through what this is," she said.