Sign-count conjecture for the global quantity on rectangular tori

Let $E=\mathbb C/\Lambda$ be a rectangular torus, let $S$ be a set of branch points on the associated hyperelliptic curve, and let $D(S)$ be the global quantity defined from the corresponding Green-function data.

Sign-count conjecture. There are $n$ branch points on the associated hyperelliptic curve for which $D(S)<0$ and $n+1$ branch points for which $D(S)>0$.

The conjecture concerns the sign distribution of the quantity $D(S)$, which controls information about bubbling solutions of the mean field equation. The supplied text gives sign computations in special cases, but does not state a resolution of this general rectangular-torus count.