The Gamma-zero-zero base-locus conjecture for Jacobians

For a principally polarized abelian variety $(A,\Theta)\in{\mathcal A}_g^{\mathrm{ind}}$, let $\Gamma_{00}$ be the linear system described in the source and let $Bs(\Gamma_{00})$ denote its base locus. The point $0$ is always in this base locus.

$\Gamma_{00}$ conjecture. If

$$Bs(\Gamma_{00})\ne\{0\},$$

then $A$ belongs to the Jacobian locus:

$$A\in{\mathcal J}_g.$$

The conjecture is intended as a characterization of Jacobians through the base locus of the $\Gamma_{00}$ system. The source notes that proving it for a generic ppav would not by itself solve the Schottky problem, since a generic ppav is not a Jacobian.