The positive-characteristic dynamical Mordell–Lang conjecture for polynomial product maps

Let $K$ be a field of characteristic $p>0$, let $f,g\in K[x]$, and define the regular self-map $\Phi:{\mathbb A}^2\longrightarrow{\mathbb A}^2$ by

$$\Phi(x,y)=\left(f(x),g(y)\right).$$

For a point $\gamma\in{\mathbb A}^2(K)$ and a curve $C\subset{\mathbb A}^2$ defined over $K$, define the return set

$$R:=\left\{n\in{\mathbb N}_0\colon\Phi^n(\gamma)\in C\right\}.$$

Positive-characteristic dynamical Mordell–Lang conjecture. The return set $R$ is a union of finitely many infinite arithmetic progressions together with finitely many sets of the form

$$\left\{ap^{rk}+b\colon k\in{\mathbb N}_0\right\},$$

for some rational numbers $a,b$ and some non-negative integer $r$.

This is a positive-characteristic special case of a broader dynamical Mordell–Lang question. It is connected to the polynomial orbit-intersection conjecture through the diagonal curve in ${\mathbb A}^2$, but the source does not report a resolution.