Boundary Andreotti–Mayer conjecture for tangentially degenerate loci

Let $(B,\Xi)$ be a simple principally polarized abelian variety of dimension $g$, and suppose that $(B,\Xi)\notin N_{g,i}$ for every $i\geq k\geq1$. Here $N_k(B,\Xi)\subset B$ is the locus of points $b$ such that $\Xi$ and its translate $\Xi_b$ are tangentially degenerate along a subvariety of dimension $k$. Let $Z$ be an irreducible component of $N_k(B,\Xi)$, and let $\operatorname{codim}_B(Z)$ denote its codimension in $B$. Boundary Andreotti–Mayer conjecture. There is such a component $Z$ with

$$\operatorname{codim}_B(Z)=k+1$$

if and only if either

$$g\geq2,\quad k=g-2,\quad\text{and}\quad B\text{ is a hyperelliptic Jacobian},$$

or

$$g\geq3,\quad k=g-3,\quad\text{and}\quad B\text{ is a Jacobian}.$$

This is proposed as a boundary version of the Andreotti–Mayer conjecture and as a conjectural characterization of Jacobians and hyperelliptic Jacobians among simple principally polarized abelian varieties; the supplied text gives no resolution.