More can be said on @StaceyStarwolf 's conjecture by simply referring to other discussions in CollatzWorld:
What is claimed is that an odd integer q is prime if and only if the above dynamical system generates cycles with an identical number of divisions by 2. Call the latter property SSW and write SSW(q) whenever it is satisfied for q.
First, SSW dynamical system can be changed onto the map x \mapsto (x+1)/2 when x odd and x \mapsto x/2 otherwise, if we apply it to rationals with denominator q. A similar change has been suggested for variants of the Collatz map of the form (3x+q)/2 vs (3x+1)/2, but I couldn’t find back a relevant thread. This is quite obvious.
In fact, this new map is the same as @Failix’s tricot 2x+1 → x+1 and 2x → x already discussed in this thread. An argument was given in this post (with the help of chatGPT) for why all cycles have the same length when starting from a rational a/q with gdc(a,q)=1 and q fixed.
So prime(q) \Rightarrow SSW(q) as claimed.
I’m pretty sure the converse can be proved as well.