Habegger–Pila optimal-subvariety conjecture for abelian varieties

Let $K$ be an algebraically closed field and let $d$ be a non-negative integer. Let $A$ be an abelian variety defined over $K$ and let $V$ be a subvariety of $A$. For a subvariety $U$ of $A$, let $\langle U\rangle$ be the smallest special subvariety containing it and define its defect by

$$\delta(U)=\dim\langle U\rangle-\dim U.$$

A subvariety $W$ of $V$ is optimal for $V$ in $A$ if $\delta(U)>\delta(W)$ for every subvariety $U$ with $W\subsetneq U\subset V$.

Habegger–Pila optimality conjecture. The subvariety $V$ contains at most finitely many optimal subvarieties of defect at most $d$.

This conjecture is stated as an equivalent formulation of the atypical-intersection conjecture above. It was formulated by Habegger and Pila, building on the notion of defect introduced by Pink; the source does not specify a resolution in this generality.