Ih's finiteness conjecture for preperiodic points of rational-function semigroups

Let $K$ be a number field and let $S$ be a finite set of places of $K$ containing all archimedean ones. Let $f_1,\ldots,f_s:\mathbb{P}^1\to\mathbb{P}^1$ be rational maps of degree at least $2$ defined over $K$, and let ${\mathcal G}:=\langle f_1,\ldots,f_s\rangle$. A point $\alpha\in\mathbb{P}^1(\overline K)$ is preperiodic for ${\mathcal G}$ if $f_{i_1\cdots i_n}(\alpha)=f_{i_1\cdots i_m}(\alpha)$ for some $n>m\geq0$ and $i_1,\ldots,i_n\in\{1,\ldots,s\}$. Semigroup Ih conjecture. If $\beta\in\mathbb{P}^1(\overline K)$ is not preperiodic for ${\mathcal G}$, then there are only finitely many preperiodic points $\alpha\in\mathbb{P}^1(\overline K)$ for ${\mathcal G}$ which are $S$-integral relative to $\beta$. This is the proposed integrality generalization for finitely generated semigroups of rational functions; the source does not provide a resolution.