The geometric Bogomolov assertion for a closed subvariety

Let $A$ be an abelian variety over $\overline{K}$ and let $X$ be a closed subvariety of $A$. The geometric Bogomolov assertion. If $X$ has dense small points, then $X$ is a special subvariety. This is the implication proved in the paper for divisors and curves, while the converse direction is part of the full geometric Bogomolov conjecture; the full assertion remains open in general.