Izadi's base-locus conjecture for Prym second-order theta divisors

Let $(P,\Xi)$ be a Prym variety associated with a double cover $\pi:\widetilde C\to C$, and let $\Sigma$ be the relevant Prym-theta subvariety. Let $\Gamma_{00}'$ consist of sections in $\Gamma_{00}$ whose restriction to $\Sigma$ vanishes, let $|2\Xi|_{00}'$ be its projectivized linear system, and let $V_{00}'$ be its base locus. Let ${\cal Q}_{00}'$ be the corresponding system of quartic tangent cones at the origin, with base locus $V_{\inf,00}'$. Let $\chi C$ denote the Prym-canonical image of $C$ in $|\omega_C\otimes\alpha|^*$, where $\alpha$ is the square-trivial invertible sheaf associated with $\pi$. Izadi's Prym base-locus conjecture. If $(P,\Xi)$ is not a Jacobian, then

1. $V_{00}'=\Sigma$ as a set if $p\geq 6$;
2. $V_{\inf,00}'=\chi C$ as a set if $p\geq 6$.

This conjecture asks whether the restricted second-order theta systems detect precisely the Prym-theta subvariety and its Prym-canonical tangent-cone curve away from the Jacobian locus. The source poses it after proving dimension and codimension bounds; no resolution is specified here.