Zilber's Exponential Algebraic Closedness conjecture for semiabelian varieties

Let $S$ be a complex semiabelian variety of dimension $n$, and write $\exp_S: \mathbb{C}^n \to S$ for its exponential map. Let $V \subseteq \mathbb{C}^n \times S$ be a free and rotund subvariety. Zilber's Exponential Algebraic Closedness conjecture for semiabelian varieties. There is a point $\bar{z} \in \mathbb{C}^n$ such that $(\bar{z},\exp_S(\bar{z})) \in V$. This extends the EAC formulation from algebraic tori to exponential maps of complex semiabelian varieties. The paper presents this as a conjecture and proves special cases when the variety has dominant projection to the domain; the general assertion remains open.