Folk conjecture on triple points of theta divisors

Let ${\mathcal A}_g$ be the moduli space of principally polarized abelian varieties. Let $T_3^{(g)}$ denote the locus of triples $(A,\Theta,x)$ where the theta divisor has multiplicity at least three at $x$, let $F_{3,0}^{(g)}=\pi(T_3^{(g)})$, and let ${\mathcal X}_g^{\rm dec}$ denote the decomposable locus in the universal principally polarized abelian variety. Write $\lfloor\frac{g+3}{2}\rfloor$ for the floor of $(g+3)/2$.

Folk conjecture on triple points.

$$T^{(g)}_{\left\lfloor\frac{g+3}{2}\right\rfloor}\subset{\mathcal X}_g^{dec}.$$

The conjecture asserts that theta divisors with multiplicity at least $\left\lfloor(g+3)/2\right\rfloor$ occur only over decomposable ppav. It is motivated by the observation that indecomposable ppav have maximal possible theta-divisor multiplicity at most $(g+1)/2$. The source gives no resolution status.