Zilber's free-and-rotund exponential-algebraic closedness conjecture

Let $V\subseteq\mathbb{C}^n\times(\mathbb{C}^{\times})^n$ be an algebraic variety, and let $\pi_1$ and $\pi_2$ be the projections to the additive and multiplicative factors. Call $V$ free if $\pi_1(V)$ is not contained in a translate of a $\mathbb{Q}$-linear subspace and $\pi_2(V)$ is not contained in a translate of an algebraic subgroup. Let $L\leq\mathbb{C}^n$ be a $\mathbb{Q}$-linear subspace, let $\operatorname{exp}(L)$ be its coordinatewise exponential image, and let

$$\pi_L:\mathbb{C}^n\times(\mathbb{C}^{\times})^n\twoheadrightarrow \mathbb{C}^n/L\times(\mathbb{C}^{\times})^n/\operatorname{exp}(L)$$

be the quotient map. Call $V$ rotund if $\dim(\pi_L(V))\geq n-\dim L$ for every such $L$. Zilber's free-and-rotund exponential-algebraic closedness conjecture. If $V$ is free and rotund, then

$$V\cap\Gamma_{\operatorname{exp}}\neq\varnothing,$$

where $\Gamma_{\operatorname{exp}}$ is the graph of the $n$-fold coordinatewise exponential map. This is the precise special form studied in the paper; the paper proves a case of the broader exponential-algebraic closedness problem, while this conjectural statement remains open.