Sokal's simple-zero conjecture for the entire function F

Let

$$F(z;q)=\sum_{k=0}^{\infty}\frac{q^{k(k-1)/2}z^k}{k!},$$

where $q\in\mathbb{C}$ satisfies $0<|q|\leq 1$. Sokal's conjecture. The entire function $F(z;q)$ can have only simple zeros. The conjecture concerns the multiplicity of the zeros of this generalized exponential function. The claim is known for positive and real $q$, but remains open in the stated complex-parameter generality according to the supplied text.