Van Geemen–van der Geer's conjecture on the \Gamma_{00} base locus

Let $(A,\Theta)$ be an indecomposable principally polarized abelian variety of dimension $g$, with $\Theta$ symmetric. Let $\Gamma_{00}\subset H^0(A,2\Theta)$ be the subspace of sections vanishing to order four at the origin, and let $V_{00}$ be the base locus of the associated linear series. For a smooth curve $C$, write

$$C-C=\{\mathscr O_C(p-q):p,q\in C\}.$$

Van Geemen–van der Geer's conjecture. If $(A,\Theta)$ is the Jacobian of a smooth curve $C$, then, as a set,

$$V_{00}=C-C.$$

If $(A,\Theta)$ is not a Jacobian, then, as a set, $V_{00}=0$. This conjecture seeks to characterize Jacobians among indecomposable principally polarized abelian varieties through the base locus of the linear system defined by sections vanishing to order four at the origin. The source gives no resolution status beyond stating the conjecture.