Codimension conjecture for the singular theta-divisor locus

Let ${\mathcal X}_g$ be the universal principally polarized abelian variety, and let $\Theta_{\rm sing}\subseteq {\mathcal X}_g$ be the locus of points lying on a theta divisor together with the vanishing of all first and second derivatives of the theta function, equivalently the relative singular locus of the theta divisor. Codimension conjecture for $\Theta_{\rm sing}$. For every $g\geq 3$,

$$\operatorname{codim}_{{\mathcal X}_g}\Theta_{\rm sing}=2g.$$

This is one of the paper's proposed codimension predictions for loci of abelian varieties with highly singular theta divisors; the supplied text gives no resolution.