Polynomial-exponential equations and Zilber's conjecture

Assuming Schanuel's conjecture, we prove that any polynomial-exponential equation in one variable must have a solution that is transcendental over a given finitely generated field. With the help of some recent results in Diophantine geometry, we obtain the result by proving (unconditionally) that certain polynomial-exponential equations have only finitely many rational solutions. This answers affirmatively a question of David Marker, who asked, and proved in the case of algebraic coefficients, whether at least the one variable case of Zilber's strong exponential-algebraic closedness conjecture can be reduced to Schanuel's conjecture.