Geometric Bogomolov conjecture for abelian varieties

Let $K$ be a function field with algebraic closure $\overline{K}$, let $A$ be an abelian variety over $\overline{K}$, and let $X$ be a closed subvariety of $A$. Say that $X$ has dense small points if, for an even ample line bundle $L$ on $A$, the set $X(\epsilon;L)$ is dense in $X$ for every $\epsilon>0$. Say that $X$ is special if there exist an abelian subvariety $G$ of $A$, a torsion point $\tau\in A(\overline{K})_{tor}$, and a closed subvariety $\widetilde{Y}\subset\widetilde{A}^{\overline{K}/k}$ such that

$$X=\operatorname{Tr}_A\left(\widetilde{Y}\otimes_k\overline{K}\right)+G+\tau.$$

Geometric Bogomolov conjecture. If $X$ has dense small points, then $X$ should be special.

This is the function-field analogue of the Bogomolov conjecture for curves, with special subvarieties replacing torsion subvarieties. The supplied text does not state a resolution status.