Zhang's dynamical Manin–Mumford conjecture

Let $X$ be a variety and let $\Phi:X\to X$ be a dominant map. A point of $X$ is preperiodic for $\Phi$ if its forward orbit under $\Phi$ is finite, and a subvariety $Y$ is preperiodic if some two iterates $\Phi^m(Y)$ and $\Phi^n(Y)$ with $m<n$ are equal. Zhang's dynamical Manin–Mumford conjecture. The subvariety $Y$ contains a Zariski dense subset of preperiodic points if and only if $Y$ is preperiodic.

This generalizes the Manin–Mumford conjecture for subvarieties of abelian varieties containing infinitely many torsion points. The status evidence supplied with the candidate says that the conjecture is a generalization of Raynaud's proved Manin–Mumford theorem, but does not establish resolution of the dynamical statement itself.