Yamaki's 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 $A$. A closed subvariety has dense small points when its points of arbitrarily small canonical height are dense, and it is special in the sense of the geometric Bogomolov conjecture. Yamaki's geometric Bogomolov conjecture. $X$ has dense small points if and only if $X$ is a special subvariety. The paper proves the conjecture for curves and divisors, and for abelian varieties of dimension at most $3$ or satisfying the stated nowhere-degeneracy and trace dimension bound; it remains open in higher-dimensional cases.