The indecomposable theta-multiplicity conjecture for principally polarized abelian varieties

Let $(A,\Theta)$ be a principally polarized abelian variety of dimension $g$, and let $S_k\subseteq {\mathcal A}_g$ be the locus of ppavs whose theta divisor has a point of multiplicity at least $k$. A ppav is indecomposable if it is not a product of lower-dimensional principally polarized abelian varieties.

Theta-multiplicity conjecture. The maximal multiplicity of the theta function for indecomposable ppavs is

$$\left\lfloor\frac{g+1}{2}\right\rfloor,$$

i.e. $S_{\lfloor\frac{g+3}{2}\rfloor}$ is contained in the locus of decomposable abelian varieties.

The conjecture is motivated by the maximal multiplicities known for Jacobians and Prym varieties and by the behavior of hyperelliptic Jacobians. The source gives no resolution status.