Beauville–Debarre infinitesimal Schottky conjecture for Prym and Jacobian theta divisors

Let $(P,\Xi)$ be a principally polarized abelian variety of dimension $p\geq 4$, and let $V_{\inf,00}$ be the base locus of the linear system of quartic tangent cones at the origin arising from sections of $\Gamma_{00}\subset H^0(P,2\Xi)$, where sections vanish to order at least $4$ at the origin. For a smooth curve $C$ of genus $g$, let $(JC,\Theta)$ be its Jacobian, let $\kappa C$ be the canonical image of $C$ in $|\omega_C|^*=\mathbb P T_0JC$, and let ${\cal J}_g$ be the Jacobian locus in the moduli space of principally polarized abelian varieties of dimension $g$. Beauville–Debarre conjecture.

1. If $(P,\Xi)=(JC,\Theta)$, then $V_{\inf,00}$ is, set-theoretically, the canonical image $\kappa C$ of $C$ in $|\omega_C|^*=\mathbb P T_0JC$.
2. If $(P,\Xi)$ is not in ${\cal J}_g$, then $V_{\inf,00}$ is empty.

The conjecture is the infinitesimal version of the preceding proposed Schottky characterization: the base locus of quartic tangent cones should recover the canonical curve for Jacobians and disappear away from the Jacobian locus. Its resolution is not specified in the source.