Relative dynamical Bogomolov conjecture for split polarized endomorphisms

Let $K$ be the base field, let $\ell\geq 2$, and let $\mathbf{F}:\mathbb{P}^{\ell}_1\to\mathbb{P}^{\ell}_1$ be an isotrivial-free split polarized endomorphism over $K$ with polarization degree at least $2$. Let $\mathbf{V}\subset\mathbb{P}^{\ell}_1$ be an irreducible projective subvariety defined over $K$ such that

$$\dim \mathbf{V}<\ell-1.$$

Assume that $\mathbf{V}$ is not contained in an $\mathbf{F}$-preperiodic subvariety other than $\mathbb{P}^{\ell}_1$. Write $\hat{h}_{F}$ for the fiber-wise height associated with $\mathbf{F}$.

Relative dynamical Bogomolov conjecture. There exists $\epsilon=\epsilon(\mathbf{F},\mathbf{V})>0$ such that

$$\left\{P\in\mathbf{V}(\overline{\mathbb{Q}}):\hat{h}_{F}(P)\leq\epsilon\right\}$$

is not Zariski dense in $\mathbf{V}$.

This is presented as a generalization suggested by the Pink–Zilber conjectures and as a dynamical analogue of the relative Bogomolov conjecture. The source states it as a proposed general principle; the paper proves particular relative and geometric Bogomolov results, but does not establish this full statement.