Equality of the compactified gradient and odd two-torsion singularity loci

For each genus $g$, let $I^{(g)}\subset\mathcal A_g$ be the locus of principally polarized abelian varieties whose theta divisor is singular at an odd two-torsion point, and let $G^{(g)}$ denote the corresponding locus defined by the vanishing of the relevant theta-gradient equations. Write $\overline{I^{(g)}}$ and $\overline{G^{(g)}}$ for their closures in the full perfect-cone compactification $\mathcal A_g^{\operatorname{Perf}}(8)$. Equality conjecture. For every genus $g$,

$$\overline{G^{(g)}}=\overline{I^{(g)}}.$$

This predicts that passing to the full compactification introduces no additional boundary component into the zero locus of the theta-gradient equations. The source presents the equality as a conjecture, with no resolution supplied.