Conjecture on the number and modular invariance of critical points of the interaction functional

Let $\mathbb{H}=\{z\in\mathbb{C}:\operatorname{Im}(z)>0\}$ be the upper half-plane, and let

$$\mathcal{J}(z;a,b)=\sum_{(m,n)\in\mathbb{Z}^2}e^{-\frac{\pi}{\operatorname{Im}(z)}|mz-n|^2}\cos(2\pi(ma+nb)).$$

For a fixed $z$, consider the critical points of $\mathcal{J}(z;a,b)$ with respect to $(a,b)$. Let $\Omega_4$ and $\Omega_6$ denote the subsets of $\mathbb{H}$ corresponding to tori $z$ with four and six such critical points, respectively. Let

\Gamma=SL_2(\mathbb{Z})=\left\{\begin{pmatrix}a&b\c&d\end{pmatrix}:ad-bc=1,\ a,b,c,d\in\mathbb{Z}\right\}.

Critical-point conjecture. The function $\mathcal{J}(z;a,b)$ has either four or six critical points with respect to $a,b$, and the following hold:

$$\mathbb{H}=\Omega_4\cup\Omega_6,\qquad \Omega_4\cap\Omega_6=\emptyset.$$

Rectangular tori have only four critical points and the hexagonal torus has six; in particular,

$$i\in\{z:\operatorname{Re}(z)=0,\ \operatorname{Im}(z)>0\}\subset\Omega_4,\qquad \frac{1}{2}+i\frac{\sqrt{3}}{2}\in\Omega_6.$$

Moreover,

$$z\in\Omega_4\Rightarrow\Gamma(z)\in\Omega_4,\qquad z\in\Omega_6\Rightarrow\Gamma(z)\in\Omega_6.$$

This conjecture concerns the global classification of critical points of the relative-position interaction and its compatibility with modular symmetry; the source says it is suggested by numerical simulation, and no resolution is supplied.