ThisThis 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.