Collatz conjecture in various numeral systems also asymmetric

There is this legendary https://en.wikipedia.org/wiki/Collatz_conjecture even getting Veritasium video "The Simplest Math Problem No One Can Solve": that using rule "divide x by 2 if even, take 3x+1 otherwise" at least experimentally from any natural number there is reached 1 - any thoughts?

It seems natural to try to look at evolution of x in numeral systems: base-2 is natural for x->x/2 rule (left column), but base-3 does not look natural for x->3x+1 rule (central column).

However, rANS ( https://en.wikipedia.org/wiki/Asymmetric_numeral_systems ) gluing 0 and 2 digits of base-3 (removing 1 bit of information) looks quite natural (right column) - maybe some rule could be found from it helping to prove this conjecture?

Edited September 24Sep 24 by Duda Jarek