Weak Complex Ax conjecture, geometric formulation

Let $U$ be a domain in $\mathbb{H}^n$, let $U'$ be a geodesic subvariety of $U$, and put $X'=\exp U'$. Let $A$ be a complex-analytic component of $W\cap\pi^{-1}(V)$, where $W\subseteq U'$ and $V\subseteq X'$ are algebraic subvarieties. Assume that $A$ is not contained in any proper geodesic subvariety of $U'$. Weak Complex Ax conjecture. Then

$$\dim A\leq\dim V+\dim W-\dim X'.$$

This is presented as a geometric formulation of the complex Ax-Schanuel principle: intersections should not have atypically large dimension unless they lie in a proper geodesic subvariety. The source states that the analogous weak Ax result holds for semi-abelian varieties by Ax's theorem, but does not resolve this modular formulation.