Is there something wrong with that proof? https://www.youtube.com/watch?v=O7SmNyte6nA (in French)
The 3x+1 Conjecture is false Text in English : https://arxiv.org/abs/2104.10681
The 3x+1 Conjecture is false Text in English : https://arxiv.org/abs/2104.10681
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.