Sokal's distinct-moduli conjecture for the disturbed exponential function

Let

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

where $q$ is a complex number satisfying $0<|q|\leq 1$. Sokal's stronger conjecture. The function $\mathcal F(z;q)$ can have only simple zeros with distinct absolute values. This strengthens the simplicity conjecture. The assertion is proved in some special cases, including purely imaginary $q$, but remains open for general complex $q$.