Pure-codimension conjecture for the odd two-torsion singularity locus

Let $\mathcal A_g$ be the moduli space of principally polarized abelian varieties of genus $g$, and let $I^{(g)}\subset\mathcal A_g$ be the locus of pairs $(A,\Theta)$ whose theta divisor is singular at some odd two-torsion point $m\in A[2]$. Pure-codimension conjecture. The locus $I^{(g)}$ has pure codimension $g$ in $\mathcal A_g$ for every $g$, and is reduced. This conjecture concerns the expected geometric and scheme-theoretic structure of the odd two-torsion singularity locus; the source states that the locus is not well understood in higher genus and presents the assertion as open.