Debarre's strong Schottky conjecture for theta divisors

Let $(A,\Theta)$ be an indecomposable principally polarized abelian variety of dimension $g$, so that $\Theta$ is irreducible. For $m\geq 2$, define

$$\mathrm{Sing}_m(\Theta)=\{x\in\Theta\mid \mathrm{mult}_x\Theta\geq m\}.$$

Debarre's strong Schottky conjecture. If $(A,\Theta)$ is not a hyperelliptic Jacobian or the intermediate Jacobian of a smooth cubic threefold, then

$$\dim \mathrm{Sing}_m(\Theta)\leq g-2m$$

for every $m\geq 2$.

This stronger bound is intended to characterize the boundary cases of the general singular-locus bound and would give a solution to the geometric Riemann–Schottky problem. Its case $m=2$ is attributed to Debarre and was proposed by Grushevsky; the general conjecture remains open.