Codimension conjecture for the theta-null gradient locus

Let ${\mathcal A}_g$ be the moduli space of principally polarized abelian varieties, let ${\mathcal X}_g$ be the universal principally polarized abelian variety, and let $(\partial\theta)_{\rm null}$ be the locus in ${\mathcal A}_g$ of ppavs whose theta function has a point of order two where the theta function and all first derivatives vanish. Let $G_0$ be the corresponding locus studied in the paper. Codimension conjecture for $(\partial\theta)_{\rm null}$. For every $g\geq 3$,

$$\operatorname{codim}_{{\mathcal A}_g}(\partial\theta)_{\rm null}=\operatorname{codim}_{{\mathcal A}_g}G_0=g.$$

The paper presents this as a natural codimension prediction; the supplied text gives no resolution.