Exponentials generate holomorphic functions on simply connected convex regions

Let $R$ be a simply connected convex region of a discrete Riemann surface, and let the associated discrete exponentials be holomorphic functions on $R$. Exponential-generation conjecture. The exponentials generate the space of holomorphic functions on $R$. This concerns the completeness of discrete exponentials as holomorphic functions on suitably simple regions; the supplied text gives no resolution or further conditions.