The geometric Bogomolov conjecture for nowhere degenerate abelian varieties

Let $A$ be a nowhere degenerate abelian variety over $\overline{K}$, meaning that its abelian variety structure has no nontrivial degenerate part, and let $X$ be a closed subvariety of $A$. A subvariety is special if it is obtained from a subvariety over the constant field by the permitted homomorphism, translation by a torsion point, and addition of its stabilizer. Geometric Bogomolov conjecture for nowhere degenerate abelian varieties. $X$ should not have dense small points unless it is a special subvariety. This formulation is equivalent to the general conjecture after reducing to the maximal nowhere degenerate abelian subvariety; the relevant general case remains open.