The positive-characteristic intersection-of-orbits conjecture for polynomials

Let $K$ be a field of characteristic $p$, and let $f,g\in K[x]$ be polynomials of degrees larger than $1$. Suppose that there exist $\alpha,\beta\in K$ such that $\mathcal{O}_f(\alpha)\cap\mathcal{O}_g(\beta)$ is infinite. A polynomial is additive if it satisfies $h(x+y)=h(x)+h(y)$, and a polynomial is linearly conjugate to $h$ if it has the form $\lambda^{-1}\circ h\circ\lambda$ for a linear polynomial $\lambda$.

Intersection-of-orbits conjecture. At least one of the following holds: (1) there exist $m,n\in{\mathbb N}$ such that $f^m=g^n$; or (2) there exist linear polynomials $\lambda,\mu\in\overline{K}[x]$ and additive polynomials $\widetilde f,\widetilde g\in\overline{K}[x]$ such that

$$f=\lambda^{-1}\circ\widetilde f\circ\lambda,\qquad g=\mu^{-1}\circ\widetilde g\circ\mu,$$

and there exists $m\in{\mathbb N}$ such that

$$\widetilde f^m\circ\widetilde g^m=\widetilde g^m\circ\widetilde f^m.$$

This conjecture modifies the common-iterate conclusion known over characteristic zero to account for positive-characteristic counterexamples involving additive polynomials. Its full generality remains open, and no counterexamples are known.