Strong semigroup version of Ih's finiteness conjecture

Let $K$ be a number field and let $S$ be a finite set of places of $K$ containing all archimedean ones. Let $f_i:\mathbb{P}^1\to\mathbb{P}^1$, $i\in I$, be rational maps of degree at least $2$ defined over $K$, and let ${\mathcal G}:=\langle f_i\rangle_{i\in I}$. A point is strongly preperiodic for ${\mathcal G}$ if its ${\mathcal G}$-orbit is finite. Strong semigroup Ih conjecture. If $\beta\in\mathbb{P}^1(\overline K)$ is not strongly preperiodic for ${\mathcal G}$, then there are only finitely many strongly preperiodic points $\alpha\in\mathbb{P}^1(\overline K)$ for ${\mathcal G}$ which are $S$-integral relative to $\beta$. This formulation is equivalent to the single-map conjecture in the paper and is vacuous when the semigroup contains dynamically unrelated elements with distinct Julia sets; the general formulation is not presented as resolved.