Good-reduction conjecture for polarized morphisms

Let $k$ be any complete non-Archimedean field, and let $X$ be a projective $k$-variety. A morphism $f\colon X\to X$ is polarized if there exists an ample line bundle $L\to X$ and an integer $q\ge 2$ such that $f^*L$ is linearly equivalent to $L^{\otimes q}$. The morphism $f$ has good reduction if it extends to a regular self-map of a suitable flat projective model of $X$ over the valuation ring of $k$. Good-reduction conjecture. If $f\colon X\to X$ is polarized and $\operatorname{h_{top}}(f)=0$, then $f$ has good reduction. This asks whether, within the polarized setting, vanishing topological entropy forces good reduction. The surrounding discussion explains that the analogous assertion fails for general regular self-maps in higher dimensions, while the polarized case is proposed as the relevant positive statement.