[morei]

I had a quick read. The paper states in the conclusion "The more we increase the number of iterations k, the more proportion decreases, but the number of integers starting the different trajectories keeps increasing and tends towards infinity as k tends towards infinity."

This statement is not proven, and is not supported by the previous theorems. Thus the following statement "We therefore conclude that the 3x+ 1 conjecture is false" remains an unproven assertion.

Essentially the paper argues that the density F(k) of trajectories at least as long as 'k' gets ever smaller as k increases, without ever reaching zero, and thus there exists at least one value that has an infinity long trajectory.

But that's as false assertion. Just because F(k) is non-zero for every finite value of k does NOT imply that the lim as k tends to infinity of F(k) is non-zero, and consequently does not prove the existence of an infinity long trajectory.

TL;DR: paper applies intuition on finite sequences to infinite sequences and gets it wrong.