Zilber's strong exponential-algebraic closedness conjecture in one variable

Let $k\subset\mathbb{C}$ be a finitely generated field, and let $p(x,y)\in k[x,y]$ be irreducible with $\frac{\partial p}{\partial x},\frac{\partial p}{\partial y}\neq 0$. Zilber's strong exponential-algebraic closedness conjecture. There exists $z\in\mathbb{C}$ such that

$$p(z,\exp(z))=0$$

and

$$\operatorname{trdeg}_{k} k(z,\exp(z))=1.$$

This is the one-variable case of Zilber's strong exponential-algebraic closedness property, which concerns solutions of polynomial-exponential systems compatible with Schanuel's conjecture. The paper proves this assertion assuming Schanuel's conjecture, while the unconditional general statement remains open.