Exact-solution-count conjecture for singular Liouville equations on rectangular tori

Let $E_\tau$ be a rectangular torus with $\tau\in i\mathbb{R}_{>0}$, let $n\in\mathbb{N}$, and let $\rho\in(8\pi(n-1),8\pi n)$. Consider

$$\Delta u+e^u=\rho\delta_0\quad\text{on }E_\tau.$$

Exact-solution-count conjecture. The equation possesses exactly $n$ solutions.

This refines the nonexistence results at the quantized values $\rho=8\pi n$ by predicting the precise number of solutions between consecutive multiples of $8\pi$. The supplied text does not state a resolution of this conjecture.