Debarre's codimension conjecture for Andreotti–Mayer loci

For $g\geq 1$, let $N_{k,g}$ be the Andreotti–Mayer locus of principally polarized abelian varieties whose theta divisor has singular locus of dimension at least $k$, let ${\mathcal A}_g^{\mathrm{ind}}$ denote the indecomposable locus, and let $Hyp_g$ be the locus of hyperelliptic Jacobians. For an irreducible component $X\subset N_{k,g}$, assume that for general $(A,\Theta)\in X$ one has $\operatorname{End}(A,\Theta)={\mathbb Z}$.

Debarre's codimension conjecture. For any $1\leq k\leq g-3$,

$$\operatorname{codim}_{{\mathcal A}_g}X\geq\frac{(k+1)(k+2)}{2},$$

with equality only for components ${\mathcal J}_g\subset N_{g-4,g}$ and $Hyp_g\subset N_{g-3,g}$. This conjecture concerns the expected size of Andreotti–Mayer loci and their role in solving the Schottky problem. The source gives no resolution status.