I mentioned the Gordon conjecture in this previous post. At the time, there were two proofs on the arXiv, niether of which had been completely confirmed. Now the story of the conjecture seems to be more or less complete, so I thought I’d give an update and account of the story (at least as far as I understand it).
The Gordon conjecture is as follows: Given two Heegaard splittings of two closed 3-manifolds, one can consider the connect sum of those 3-manifolds: remove an open ball from each manifold and glue the two together along the sphere boundary components that result. If one chooses the balls to intersect the Heegaard surfaces in the right way and chooses the right gluing map, the images of the two Heegaard surfaces will induce a Heegaard splitting for the connect sum. This Heegaard surface is called the connect sum of the original two Heegaard surfaces. The Gordon conjecture states that if the original two Heegaard surfaces are unstabilized then the connect sum Heegaard surface is also unstabilized.
Dave Bachman posted a preprint  on the arXiv in 2004 (so about four years ago) claiming to prove this conjecture. This attracted a lot of interest at first, but the paper and the proof both proved difficult for the rest of the Heegaard splitting community to understand. It is based on machinery called sequences of generalized Heegaard splittings, which I wrote a few posts on here, here and here. However, because of the difficulties in reading the paper, the proof was never really accepted and many considered the problem still open.
Luckily, the editors of GT (where Dave submitted the paper) found a referee who was willing to put a huge amount of time into suggesting ways to improve the exposition of the paper. The final version (which doesn’t seem to be on the arXiv yet) was accepted into GT just recently. After writing the three posts on SOGs, I think I understand the proof well enough to believe it. (Reading those posts may not be as helpful as it was to write them; I never got around to writing about the final step in the proof.) So four years later, Dave’s proof seems to be confirmed.
Now, a few months after Dave’s paper appeared, another paper  claiming to prove Gordon’s conjecture was posted on the arXiv by Ruifeng Qiu. This paper also proved to be almost entirely unreadable. This paper also sat in limbo for a number of years, with no one willing to confirm or deny its accuracy. Finally, a year or so ago, Qiu invited Martin Scharlemann to China to discuss the proof with him. After spending a number of weeks there, Marty was able to write his own account of Qiu’s proof . The paper was posted this January with only Marty’s name on it (and an explanation that this was an expository account of Qiu’s work). Earlier this month, the paper was reposted with both Scharlemann’s and Qiu’s names on it, suggesting that this will be the final paper that will be submitted for publication.
Qiu’s proof is very different from Bachman’s. Bachman’s methods are a logical extension of existing techniques, Qiu’s machinery seems to be completely new. I don’t understand the proof, but I’ve talked to reputable sources who do understand it (not to mention that Marty has an excellent track record) so Qiu’s proof seems to be in the final stages of confirmation.
So, one conjecture, two proofs, both on roughly the same time line. A messy story, indeed, but it seems to be (almost) over, and with a happy ending, no less.