The backward orbit conjecture for the powering map over global fields

Let $K$ be a finite extension of $\mathbb{F}_p(t)$, let $\phi:\mathbb{P}^1\to\mathbb{P}^1$ be a rational map of degree $d\geq 2$ defined over $K$, and let $\overline{K}$ be the separable algebraic closure of $K$. For $\beta\in\mathbb{P}^1$, write

$$\phi^-(\beta)=\bigcup_{n\geq 0}\phi^{-n}(\beta)\subset\mathbb{P}^1(\overline{K}).$$

A point $\beta$ is $\phi$-preperiodic if its forward orbit is finite. Backward orbit conjecture. If $\alpha\in\mathbb{P}^1(K)$ is not $\phi$-preperiodic, then for any $\beta\in\mathbb{P}^1(K)$, the backward orbit $\phi^-(\beta)$ contains at most finitely many points in $\mathbb{P}^1(\overline{K})$ which are $S$-integral relative to $\alpha$. The conjecture concerns the finiteness of $S$-integral points in backward orbits; the paper proves it for the powering map over a function field with finite field of constants when the exponent is relatively prime to the characteristic, while the general statement remains open.