Jones's eventual stability conjecture

Let $K$ be a number field, let $\phi\in K(x)$, and let $\alpha\in K$. The pair $(\phi,\alpha)$ is eventually stable when, writing $\phi^n=p_n/q_n$ with coprime $p_n,q_n\in K[x]$, the number of irreducible factors of $p_n-\alpha q_n$ is eventually constant. A point $\alpha$ is periodic for $\phi$ if $\phi^m(\alpha)=\alpha$ for some integer $m\ge 1$. Jones's eventual stability conjecture. The pair $(\phi,\alpha)$ is eventually stable if and only if $\alpha$ is not periodic for $\phi$. This conjecture asserts that eventual stability is the generic condition for rational maps over number fields, while the preceding remark explains its relation to stability of pairs above points in a backward orbit.