I will look at it this weekend. If true, the productivity of math departments everywhere will go up. I know so many that banged their heads on this one for a while. I once wasted an entire week playing with this on the Symbolics. It's horribly addicting because of it's simplicity.

Fwiw, the author of the paper did his PhD work with Collatz himself, though that was over 40 years ago (http://genealogy.math.ndsu.nodak.edu/id.php?id=27958). Not conclusive, but makes me inclined to consider it a serious effort. Not qualified to judge beyond that, though.

I can't say, I looked at this briefly and was hoping for an elementary proof. This paper casts the problem in terms of linear operators over holomorphic function spaces and assumes a number of prior results in that area, and it's been far too long since I read complex analysis.

In some sense it's a little unsatisfying that a simple proof has not been found, but this is what makes math so addicting I suppose.

I'm not really a mathematician yet, but I skimmed the paper and the approach seemed reasonable when taken at face value.

Then again, I said the same thing when Penny Smith published her paper on NSE.