Ih's dynamical $S$-integral preperiodic points conjecture

Let $K$ be a number field, let $S$ be a finite set of places containing all the archimedean ones, let $f\colon \mathbb{P}^1\to\mathbb{P}^1$ be a rational function of degree at least two defined over $K$, and let $\alpha\in\mathbb{P}^1(\overline K)$ be a point that is not preperiodic for $f$. A point $\beta\in\mathbb{P}^1(\overline K)$ is $S$-integral relative to $\alpha$ if $\alpha$ and $\beta$ have distinct reduction at every place of $\overline K$ lying over a place of $K$ outside $S$. Ih's dynamical conjecture. There are at most finitely many preperiodic points $\beta\in\mathbb{P}^1(\overline K)$ that are $S$-integral relative to $\alpha$. This is a dynamical analogue of conjectures on $S$-integral torsion points relative to divisors on abelian varieties; the statement concerns a non-preperiodic reference point and a rational map of degree at least two.