The indecomposable theta-multiplicity conjecture

Let $S_k$ be the locus of principally polarized abelian varieties $(A,\Theta)$ for which there exists $z\in A$ with $\operatorname{mult}_z\Theta\geq k$, and let ${\mathcal A}_g^{\mathrm{ind}}$ be the indecomposable locus.

Theta-multiplicity conjecture.

$${\mathcal A}_g^{\mathrm{ind}}\cap S_{\left\lfloor\frac{g+3}{2}\right\rfloor}=\emptyset.$$

The conjecture reflects the expected bounds on theta-divisor multiplicities for Jacobians and Prym varieties. The source gives no resolution status.