Philippon's zero-height conjecture for subvarieties in polarized dynamical systems

Let $(X,f,\mathscr L)$ be a polarized dynamical system defined over a number field, and let $Y$ be a subvariety of $X$ with canonical subvariety height $\hat h_{\mathscr L}(Y)$. A subvariety $Y$ is preperiodic when there exist integers $n$ and $m>0$ such that $f^n(Y)=f^{n+m}(Y)$. Philippon's zero-height conjecture. If

$$\hat h_{\mathscr L}(Y)=0,$$

then $Y$ is preperiodic. Preperiodic subvarieties have canonical height zero, so the conjecture is the converse implication. The source attributes the question to Philippon and states that it is equivalent to the dynamical Bogomolov conjecture; it remains open in general, despite the known cases for abelian and related systems.