Ih's finiteness conjecture for S-integral preperiodic points

Let $K$ be a number field, let $S$ be a finite set of places of $K$ containing all archimedean places, and let $f:\mathbb{P}^1\to\mathbb{P}^1$ be a rational map of degree at least $2$ defined over $K$. A point $\alpha\in\mathbb{P}^1(\overline K)$ is $S$-integral relative to $\beta\in\mathbb{P}^1(\overline K)$ if no conjugate of $\alpha$ meets any conjugate of $\beta$ at primes outside $S$. A point is preperiodic for $f$ if $f^{n+m}(\alpha)=f^m(\alpha)$ for some $n\geq1$ and $m\geq0$. Ih's conjecture. If $\beta\in\mathbb{P}^1(\overline K)$ is not preperiodic for $f$, then there are only finitely many preperiodic points $\alpha\in\mathbb{P}^1(\overline K)$ for $f$ that are $S$-integral relative to $\beta$. This is a dynamical analogue of finiteness results for integral points and is known under certain local conditions on the non-preperiodic point $\beta$, but remains open for a general rational map.