Ciliberto–van der Geer's codimension conjecture for Andreotti–Mayer loci

Let $\mathcal A_g$ be the moduli space of principally polarized abelian varieties of dimension $g$, and let $N_{g,\ell}\subset\mathcal A_g$ be the Andreotti–Mayer locus whose theta divisor has singular locus of dimension at least $\ell$. Let $N$ be an irreducible component of $N_{g,\ell}$ whose general point corresponds to an abelian variety with endomorphism ring $\mathbb Z$.

Ciliberto–van der Geer's conjecture. If $1\le \ell\le g-3$, then

$$\operatorname{codim}_{\mathcal A_g}(N)\ge \binom{\ell+2}{2}.$$

Moreover, equality holds if and only if one of the following holds:

1. $g=\ell+3$ and $N=\bar{J}^h_g$;
2. $g=\ell+4$ and $N=\bar{J}_g$.

Here $\bar{J}^h_g$ and $\bar{J}_g$ denote the Jacobian loci occurring in the source's notation. The conjecture sharpens known lower bounds for the codimension of Andreotti–Mayer loci; the source states that it has been proved in the case $\ell=1$.