Sokal's simplicity problem for the partial theta function

Let

$$\Theta_0(z;q)=\sum_{n=0}^\infty q^{\frac{n(n-1)}2}z^n,$$

where $\widetilde q\approx 0.3092493386$ is the positive threshold described in the source. Sokal's partial-theta problem. Is it true that all zeros of $\Theta_0(z;q)$ remain simple within the open disk $|q|<\widetilde q$? The source gives partial confirmation, proving simplicity and distinct absolute values for purely imaginary parameters in a smaller range, but the full disk question remains open.