Abelian Exponential-Algebraic Closedness conjecture

Let $A$ be a complex abelian variety of dimension $g$, with exponential map

$$\operatorname{exp}_A:\mathbb{C}^g\twoheadrightarrow A,$$

and let $\Gamma_{\operatorname{exp}_A}$ be its graph. A subvariety $V\subseteq\mathbb{C}^g\times A$ is free and rotund as defined in the source: freeness excludes containment of either projection in the relevant translates of non-trivial abelian subvarieties, and rotundity requires $\dim \pi_B(V)\geq\dim(A/B)$ for every abelian subvariety $B\leq A$.

Abelian Exponential-Algebraic Closedness conjecture. If $V$ is a free and rotund algebraic subvariety of $\mathbb{C}^g\times A$, then

$$V\cap\Gamma_{\operatorname{exp}_A}\neq\varnothing.$$

This is the abelian-variety analogue of Exponential-Algebraic Closedness for the complex exponential function. It is intended to imply quasiminimality for structures arising from abelian-variety exponentials, but the source says that the statement is implicit in Bays and Kirby rather than established here.