Salvati Manni–Grushevsky codimension conjecture for triple-point loci

Let ${\mathcal A}_g$ be the moduli space of principally polarized abelian varieties, let $T_3^{(g)}$ be the locus of ppav together with a point where the theta divisor has multiplicity at least three, and let $T_3^{(g)}[2]$ be the corresponding locus with the point restricted to a $2$-torsion point. Let $\pi$ forget the marked point, and define

$$F_{3,0}^{(g)}=\pi(T_3^{(g)}),\qquad I^{(g)}:=\pi(T_3^{(g)}[2]).$$

Salvati Manni–Grushevsky codimension conjecture. The loci $F_{3,0}^{(g)}$ and $I^{(g)}$ are purely of codimension $g$ in ${\mathcal A}_g$.

The conjecture gives the expected codimension of the ordinary and odd two-torsion triple-point loci. The source attributes it to Salvati Manni and the first author and gives no resolution status.