Andreotti–Mayer codimension conjecture for general-endomorphism components

Let $g$ and $k$ be integers with $1\leq k\leq g-3$, and let $N$ be an irreducible component of the Andreotti–Mayer locus $N_{g,k}$ whose general point corresponds to a principally polarized abelian variety with endomorphism ring $\mathbb Z$. The notation $N_{g,k}$ denotes the locus of principally polarized abelian varieties of dimension $g$ whose theta divisor has singular locus of dimension at least $k$; $\operatorname{codim}_{{\mathcal A}_g}(N)$ is the codimension in the moduli space of principally polarized abelian varieties. Let ${\mathcal H}_g$ and ${\mathcal J}_g$ denote respectively the hyperelliptic and Jacobian loci in ${\mathcal A}_g$. Andreotti–Mayer codimension conjecture.

$$\operatorname{codim}_{{\mathcal A}_g}(N)\geq\binom{k+2}{2}.$$

Moreover, equality holds if and only if either

$$g=k+3\quad\text{and}\quad N={\mathcal H}_g,$$

or

$$g=k+4\quad\text{and}\quad N={\mathcal J}_g.$$

The conjecture is presented as the expected sharp codimension bound for Andreotti–Mayer loci and is proved in the paper only in the case $k=1$; the supplied text gives no resolution in general.