Casalaina-Martin's singular-locus conjecture for theta divisors

Let $A$ be a principally polarized abelian variety of dimension $g\geq 1$, and let $\Theta$ be a symmetric irreducible theta divisor. Equivalently, $(A,\Theta)$ is an indecomposable principally polarized abelian variety. For $m\geq 2$, define

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

Casalaina-Martin's conjecture. If $(A,\Theta)$ is an indecomposable principally polarized abelian variety, then

$$\dim \mathrm{Sing}_m(\Theta)\leq g-2m+1$$

for every $m\geq 2$.

The conjecture holds for theta divisors on Jacobians of smooth projective curves and for Prym theta divisors associated to étale double covers; in particular, it holds when $\dim A\leq 5$. The general case remains open.