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on Termination of GF(2) factor-encoding product orbits

The statement

Encode $n$ as a polynomial over $\operatorname{GF}(2)$ using its binary digits. Factor that polynomial into irreducibles, encode each factor as an integer, and multiply those encodings by ordinary integer multiplication to obtain $T(n)=A234742(n)$ . Starting from each integer absent from the image of $T$ , must iteration of $T$ reach a fixed point?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.