The dynamical Bogomolov conjecture for 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 $X$. Let the maximal preperiodic subvarieties contained in $Y$ be the finitely many maximal elements, and let $Y^*$ be their complement in $Y$. Dynamical Bogomolov conjecture. There are only finitely many maximal preperiodic subvarieties contained in $Y$, and

$$\inf_{x\in Y^*(\overline{\mathbb Q})}\hat h_{\mathscr L}(x)>0.$$

The assertion says that arbitrarily small canonical heights on a subvariety arise only from its preperiodic subvarieties. The source records proofs for toric, abelian, certain semi-abelian, quotient, and some separated-variable cases, but leaves the general polarized dynamical-system statement open.