I have two AT44 rifles, a .25 long and a .177 short, both standard (not QE) versions, and both have the same problem. With the .177, I can shoot 4 decent 5 shot groups at 29 yards, and then pump the rifle up and, without doing anything else, the next 4 groups would be a total mess. Or, with either rifle, I can shoot a pretty tight group, and then the next group will be tight, too, but in a completely different spot. I hope you find a fix, because I haven't been able to.
The stocks on both of my rifles are synthetic, and I think that flexing with the smallest movement or change of hand position is the culprit, but I may be wrong. Also engineering that allows tiny changes of pressure on the stock to cause significant changes in POI. What kind of stock does your rifle have?
I have spent hundreds of dollars and hundreds of hours screwing around with these rifles, and the only thing I have realized for my efforts so far is a mountain of frustration. Just FYI, I also corresponded with another person on the GTA forum who at one time or other had 3 different AT44s, and all had POI shift issues, whether wood stock or synthetic. So I don't think this is an uncommon problem, but it's one that the "reviewers" never mention. Strange, isn't it?