Conjecture on the zeros of the half-integral weight Eisenstein series on $\Gamma_0(4)$

Let $E_\infty(z)$ be the Eisenstein series associated to the cusp $\infty$ for the group $\Gamma_0(4)$, and let $F_\infty$ be its fundamental domain, partitioned into the regions $N$, $B$, and $P$. The weight is $k$, with $k$ in the setting of the half-integral weight Eisenstein series considered here.

Zero-location conjecture. All but one of the zeroes of $E_\infty(z)$ in $F_\infty$ lie on the circle

$$|z-\tfrac{1}{4}|=\tfrac{1}{4},$$

which is contained in the regions $N$ and $B$. The remaining zero lies in $P$ on the line $x=0$.

This conjecture predicts that the zeros lie on the boundaries or distinguished geometric loci of the relevant fundamental-domain regions, extending the observed boundary-location phenomenon for Eisenstein-series zeros. The supplied text gives no resolution, so its status is open.