Update: 2 Reflection see at end
Update: 1 I have just edited the title of the question and created a new question where I have added my given answer.
This question asks in the title for
$\qquad \qquad$ "what is the longest cycle in the Collatz ruled out so far" (1).
However, the answers which are honoured are answers for the rather opposite question
$\qquad \qquad$ "What is the currently largest known lower bound for nontrivial cycle?"(2)
The likely misformulation was indicated towards the asker in a couple of comments, and because his name is "Learner" and has low profile, one would take in account, that this might be a neglectable misnoming of the problem.
Now the OP even assigned a bounty towards an answer, which answers the (2) version of the question and makes sure that this was also his intention.
I had already deleted an answer of mine (because of deletion now visible only to $\gt 10k$ people) which made explicite the obvious misunderstanding in the title to the asker, which the asker seemingly had not been able to reflect correctly.
However, I invested much in answering for the current formulation of the title (1) and, moreover, found it now an interesting problem in itself, never been handled before this way (at least I'm not aware of such a discussion).
I've been reluctant to simply edit/adapt the title according to my interpretation of things, but tend now to do so. I think to also create an own question asking for the "currently longest explicitely disproved nontrivial cycle of general type" (except the "$m$-cycles" with low $m$ which are disproved by Steiner/Simons/deWeger for any length)
I ask here because such a rigorous reformulation (instead of simply typo-correction etc) might be felt much rude by the asker.
What do you think? Editing the title of the OP, create an own question with appropriate title to which my answer is directed and move my answer towards that new question?
Possibly the misconception of question and answers has been expressed more clearly in the comments to my first answer which was but downvoted and was not reflected properly, and which I had thus deleted for being not welcomed.
At rereading my first (and deleted) answer and the comments there again, it seems now that the problem has been with understanding the "k-cycle" concept (which would likely better called "k-peak-cycle" btw.) in difference to some "N-step-cycle", where "N-step" means a length of the cycle and "k-peak-cycle" means "hilliness" of a cycling trajectory, or how many local maxima are existent, independently of any length of that cycle in question.
See here the screenshot of the comments:
Likely the "Learner" changed his focus of question after this, not catching the meaning of the explanations at all, and adapted the interpretations of the other answerers as his original intention.
So maybe the whole process of discussing, editing, discussing it here is a double ore triple misunderstanding.
Anyway, the asker didn't seem to really communicate further and I think any more change by my part would be obsolete here.