The Exponential Closedness conjecture

Let $V \subseteq \mathbb{C}^n \times (\mathbb{C}^{\times})^n$ be an algebraic subvariety. An exponential point of $V$ is a point of the form $(z_1,\ldots,z_n,e^{z_1},\ldots,e^{z_n})$. The subvariety $V$ is called free if it has no vertical or horizontal projections, and rotund if for every $\mathbb{Q}$-defined linear subspace $Q \leq \mathbb{C}^n$, one has $\dim \pi_Q(V) \geq n-\dim Q$. Exponential Closedness conjecture. If $V$ is free and rotund, then $V$ contains an exponential point. This is a central open problem concerning the existence of solutions to systems of algebraic equations involving exponentials. It is known in dimension $n=1$ and in several further cases, including subvarieties of $\mathbb{C}^2 \times (\mathbb{C}^{\times})^2$, but remains open in general.