Short (non-) answer: I don't know.
Long (too long for a comment, so also a non-) answer: Because of how this dynamic is composed of two others, namely affine (x goes to b+ax) and divisor (greatest prime, to be precise), I am guessing the answer is yes. This is because the down step scaling appears larger than the up step scaling (citation needed).
Note that repeated applications of (b+ax) yield for initial integer x either a fixed point or a composite integer. Further, there is a constant upper bound on the number of iterations needed to reach this composite, and my guess is that the expected ratio of composite/gpf(composite) is not bounded. In the Collatz dynamic, I suspect the ratio of up to expected down scaling is not small enough to resolve the problem (citation really needed), whereas in this problem I think it can help resolve the situation.
Gerhard "Not Arithmetic Dynamicist Yet (Self-citation)" Paseman, 2017.07.10.