Benedetto–Ih finiteness conjecture for S-integral post-critically finite parameters

Let $K$ be a number field, let $S$ be a finite set of places of $K$ containing all archimedean places, and let $\beta$ be $S$-integral relative to $\beta'\in\overline{K}$ when no conjugate of $\beta$ meets any conjugate of $\beta'$ in the same residue class at any place outside $S$. Let $f_{d,c}$ denote the degree-$d$ polynomial family considered in the source, and let $\alpha\in\overline{K}$.

Benedetto–Ih conjecture. If $f_{d,\alpha}$ is not post-critically finite, then there are finitely many parameters $c\in\overline{K}$ that are $S$-integral relative to $(\alpha)$ and for which $f_{d,c}$ is post-critically finite.

This conjecture asserts a finiteness property for post-critically finite parameters satisfying an arithmetic integrality condition relative to a non-post-critically finite parameter. Its status is not determined by the supplied material.