The backward orbit integrality conjecture for Lattés maps

Let $K$ be a number field, let $S$ be a finite set of places of $K$ containing all archimedean places, and let $\phi:\mathbb P^1\to\mathbb P^1$ be a rational map of degree at least $2$ defined over $K$. For $P,Q\in\mathbb P^1(\overline K)$, write

$$\phi^-(P)=\bigcup_{n\geq 0}\phi^{-n}(P)$$

for the backward orbit of $P$, and call $Q$ $\phi$-preperiodic if its forward orbit under $\phi$ is finite.

Backward orbit integrality conjecture. If $Q\in\mathbb P^1(\overline K)$ is not $\phi$-preperiodic, then $\phi^-(P)$ contains at most finitely many points in $\mathbb P^1(\overline K)$ which are $S$-integral relative to $Q$.

This conjecture predicts finiteness of $S$-integral points in backward orbits, extending the corresponding finiteness result for forward orbits. The paper's abstract and introduction state that the conjecture is proved for Lattés maps.