The theta-function vanishing conjecture for tetrahedrally symmetric monopoles

Let $H(y)=h_{-1}(y)h_0(y)h_1(y)$ be the elliptic function defined from the functions $h_k$ in Proposition preceding the conjecture. For relatively prime integers $(m,n)=1$ satisfying $(2n-m)(n+m)>0$, set

$$y=y(\lambda)=\lambda(n+m)\rho/3,\qquad \mathcal{T}=2i\sqrt{3}(n+m)/(2n-m).$$

Theta-function vanishing conjecture. For each such pair $(m,n)$, $H(y)$ vanishes $2(|n|-1)$ times on the interval $\lambda\in(0,2)$.

This conjecture concerns the final vanishing condition needed to establish the existence of monopoles with the specified tetrahedrally symmetric spectral curve. The supplied text reports numerical evidence, but gives no resolution, so the conjecture remains open.