Intermediate-dimensional geometric Bogomolov conjecture for nowhere degenerate abelian varieties

Let $K$ be a function field with algebraic closure $\overline{K}$, and let $A$ be a nowhere degenerate abelian variety over $\overline{K}$ with trivial $\overline{K}/k$-trace. Let $X$ be a closed subvariety of $A$ satisfying

$$2\leq\dim(X)\leq\dim(A)-2.$$

Assume that $X$ has dense small points.

Geometric Bogomolov conjecture in the remaining case. Then $X$ is a torsion subvariety.

The source explicitly says that this case remains open after known results covering other dimensions and trace conditions. Here “torsion subvariety” means a translate of an abelian subvariety by a torsion point.