IsoPrymian characterization by the two-dimensional Schrödinger equation

Let $X$ be a principally polarized Abelian variety with theta function $\theta$. Let $\widehat{\theta}_1$ denote the Prym theta function associated with the first involution, let $A,U_1,U_2\in\mathbb{C}^h$ and $p_1,p_2,C\in\mathbb{C}$ be constants, and let $z$ and $\overline{z}$ be the complex variables. An isoPrymian characterization conjecture. $X$ is the isoPrymian of a smooth curve with a pair of commuting involutions, one of which has four fixed points and the other of which leaves all holomorphic Prym differentials of the first involution invariant, if and only if, for each $Z$ in an $(h-1)$-dimensional subvariety of $X$, the function

$$\psi=\frac{\widehat{\theta}_1(A+U_1z+U_2\overline{z}+Z)}{\widehat{\theta}_1(U_1z+U_2\overline{z}+Z)}\,e^{p_1z+p_2\overline{z}}$$

satisfies

$$(\partial\overline{\partial}+u)\psi=0,$$

where

$$u=2\partial\overline{\partial}\ln\theta(U_1z+U_2\overline{z}+Z)+C.$$

This is intended as a Prym analogue of Krichever's main theorem, characterizing isoPrymians through finite-gap solutions of a two-dimensional Schrödinger equation. The parser supplies no evidence that the conjecture has been resolved.