There are many posts discussing his claims and why his claims are invalid. His claims were rejected by the CCRC. The defense won't release the actual documents from the CCRC that rejects it so we can't see the reasoning in detail. The defense doesn't want anyone to see just how pathetic their claims were so won't release either CCRC decision (there was an initial decision then the final one after the defense was able to respond to the initial.
The defense appealed the CCRC rejection to the Court of Appeals and the Court of Appeals affirmed the CCRC's rejection. That decision provides a few clues as to why the CCRC rejected the claims. The CCRC rejected the claims as unsubstantiated speculation. Furthermore:

So on top of being unsubstantiated speculation the above notes the CCRC noted the speculation didn't even deal with the evidence used at trial so didn't refute the evidence that the jury's decision was based on.
So Fowler ignored the fact that blood was not found in the rifle and had the moderator not been attached then the blood would have been in the rifle. Fowler did nothing to deal with this problem he simply ignored it.
Moreover, Fowler failed to deal with the evidence of the fight in the kitchen which resulted in the moderator breaking the ceiling lamp and scratching the mantle thus getting paint on the moderator.
Here again is what Fowler's claims were and why such was rejected as unsubstantiated speculation:
Fowler's claim was that Vanezis, Fletcher and Hayward made several mistakes in relation to the non-fatal wound. 1) he says they failed to notice a muzzle imprint around the non-fatal wound. He says in his opinion they mistook a muzzle imprint as a bullet abrasion collar. Did he physically inspect the body? No. Were there any closeup photos that he could provide to the CCRC where he could point out a muzzle imprint could be seen? No? So where did his claim come from then? He decided that Vanezis' description sounds to him like Vanezis saw a muzzle imprint even though Vanezis clearly believed he didn't find any muzzle imprint.
He noted Vanezis described the wound as follows:
"The lower wound was situated on the right side of the neck, 2" below the upper wound and slightly more later. It measured 1/4" in diameter, had slight bruising at the edges as well as apparent residue marks. The surrounding skin area was also bruised."
What Vanezis et al said was being described was a bullet abrasion collar(surrounding skin to the entrance wound bruised) and stippling and sooting around the wound (residue marks)
But Fowler declared the above sounds to him like Vanezis was describing a muzzle imprint and in his opinion the imprint is consistent with the rifle muzzle sans moderator.
So he speculated:
a) that Vanezis observed a muzzle imprint though Vanezis didn't ever claim such and clearly didn't believe he saw a muzzle imprint of any kind.
b) speculated the pattern could only match the rifle sans moderator though he had no measurements to go by for the size of the so called muzzle imprint since Vanezis didn't provide any measurements for a muzzle imprint because he observed no muzzle imprint.
2) By implication he was asserting the experts were all wrong about it being a near contact wound. The only way for there to be a muzzle imprint is if there is a hard contact wound. The experts didn't find the wound to be a contact wound but rather a near contact wound. They relied on several things to make such finding including the stippling and sooting around the wound. Fowler doesn't deal with any of this he just speculates they were all wrong and that it must have actually have been a hard contact wound.
Vanezis felt he observed a near contact wound with a bullet abrasion collar. The prosecution's lab experts agreed. Fowler decided that the description provided of the collar sounds to him like a muzzle imprint and one that would match the rifle muzzle sans moderator but had zero evidence to substantiate it was a muzzle imprint and that Vanezis et al were wrong about it being a bullet abrasion collar. He had no proof that it was a hard contact wound and that they were wrong in their assessment it was a near contact wound. It was just unsupported speculation on his part.
The bottom line is that he speculated Vanezis commited an error and failed to recognize there was a muzzle imprint, speculated the muzzle imprint was consistent with the rifle sans moderator and completely ignored the problems of the lack of blood in the rifle and problems of blood and paint being found on/in the moderator.
If the non-fatal wound were a hard contact wound fired sans moderator then this wound would have resulted in blood getting inside the rifle so both wounds to Sheila would have deposited blood inside the rifle not just the second if his claims were true.
Given that the CCRC rejected his claims and that we can see how his claims were totally unsubstantiated and as the Court of Appeals put it failed to grapple with the blood and paint evidence one has to wonder why Jeremy supporters keep bringing him up.