What is the status of the Gauss Circle Problem?

When Cappell visited UWM a few months ago, one of my colleagues asked him about the status of the paper. The answer was that it is "still in works", which in plain English, probably, means "having severe problems with some remote hope to fix them". The point is that it contains no idea that hadn't been well-known to experts before it appeared, just an enormous amount of "brute forcing" (which, by the way, makes it very hard to read). Sometimes you can succeed by being just more persistent than others but it doesn't seem to be the case here. The concensus is that the existing methods have already been brought to their extreme and to proceed some fresh idea is required.

This recent arxiv posting by Shaneson claims that one may take $\epsilon > 1153/9750$, improving on Huxley's bound. It also includes the passage

In 2007 Cappell and the author posted a paper on the arXiv claiming to obtain the estimate [in my notation -- PLC] $O(r^{1/2+\epsilon})$. Unfortunately we have not been able to produce an error free version. The present paper shares with that paper the Proposition in section 5 and there is also something there akin to what immediately follows the Proposition.

I guess that's that.

Back in 2007 or so, at a tea I heard a noted expert in the field pooh-poohing it (for instance, sign errors in the Stokes analogue), and he seemed not to want to read any more re-hashes (he had seen more than one from these authors, who seemed to keep changing the argument). This expert is one of those they thank. It was unclear whether he thought their whole idea (to the extent the Intro explained this) was even capable of working. I do not know its submission status.