The uniqueness conjecture for spectral values of the partial theta function in the half-disk

Let the partial theta function be

$$\theta(q,z)=\sum_{j=0}^{\infty}q^{j(j+1)/2}z^j.$$

A spectral value is a nonzero value of $q$ for which $\theta(q,\cdot)$ has a multiple zero. The distinguished spectral values are

$$\tilde{q}_1=0.3092493386\ldots,$$

and

$$v_{\pm}=0.4353184958\ldots\pm i\,0.1230440086\ldots.$$

Uniqueness conjecture. The spectral values $\tilde{q}_1$ and $v_{\pm}$ are the only spectral values of $\theta$ for

$$|q|\leq \frac12.$$

The preceding proposition establishes that these three values are spectral values; the conjecture asserts that no others occur in the closed half-disk. The paper presents this as a conjecture motivated by the analysis in the section, while the claimed uniqueness remains unresolved there.