Everywhere eventual stability conjecture

Let $K$ be a field, let $\phi\in K(z)$ have degree $d\geq 2$, and let $\alpha\in\mathbb{P}^1(K)$ be not periodic under $\phi$. For a point $\alpha\ne\infty$, write $(\phi,\alpha)$ as eventually stable over $K$ if, for coprime $f_n,g_n\in K[z]$ satisfying $\phi^n(z)=f_n(z)/g_n(z)$, the number of irreducible factors of $f_n(z)-\alpha g_n(z)$ in $K[z]$, counted with multiplicity, is bounded independently of $n$; for $\alpha=\infty$, use the number of irreducible factors of $g_n(z)$. If $K$ is a number field, then $(\phi,\alpha)$ is eventually stable over $K$. If $K$ is a function field and $(\phi,\alpha)$ is not isotrivial, meaning that after conjugation by some $\mu\in\operatorname{PGL}_2(\overline K)$ both $\phi$ and $\alpha$ are defined over the field of constants, then $(\phi,\alpha)$ is eventually stable over $K$. Everywhere eventual stability conjecture. This holds for every number field and for every non-isotrivial pair over a function field. The conjecture extends known eventual-stability results and excludes the isotrivial function-field cases, where nonperiodic points need not exist under the relevant hypotheses.