Nonexistence conjecture for four-source Liouville equations on rectangular tori

Let $E_\tau$ be the rectangular torus with $\tau\in i\mathbb{R}_{>0}$, and let $n_0,n_1,n_2,n_3\in\mathbb{Z}_{\geq0}$. Consider the four-source equation

$$\Delta u+e^u=8\pi\sum_{k=0}^3 n_k\delta_{\omega_k/2}\quad\text{on }E_\tau,$$

where the source points are the four half-periods. Define the conditions

$$\frac{n_1+n_2-n_0-n_3}{2}\geq1,\quad n_1\geq1,\quad n_2\geq1,$$

and

$$\frac{n_1+n_2-n_0-n_3}{2}\leq-1,\quad n_0\geq1,\quad n_3\geq1.$$

Four-source nonexistence conjecture. The equation has no solution for any $\tau\in i\mathbb{R}_{>0}$ if and only if $(n_0,n_1,n_2,n_3)$ satisfies neither of these two conditions.

The conjecture removes the evenness assumption from the paper's sharp nonexistence theorem. The supplied text does not report a proof or disproof, so the conjecture remains open.