Möller's conjecture on generalized Collatz maps

Fix coprime integers $m,d>1$ with $m>d$, and define $T_{\mathrm{Möller}}:\mathbb{Z}\to\mathbb{Z}$ by the residue-class formula given in the source. Möller's conjecture. The map $T_{\mathrm{Möller}}$ eventually iterates every $x\in\mathbb{Z}$ if and only if

$$m<d^{d/(d-1)}.$$

Regardless of whether this inequality holds, $T_{\mathrm{Möller}}$ has finitely many cycles. The claim is a proposed classification of generalized Collatz dynamics and remains open in the source.