Codimension conjecture for the loci $G_k$

Let ${\mathcal A}_g$ be the moduli space of principally polarized abelian varieties, and let $G_k$ denote the loci introduced in the paper in analogy with the Andreotti–Mayer loci. Codimension conjecture for $G_k$. For every $g\geq 3$,

$$\operatorname{codim} G_k\geq g+k.$$

This is proposed as an analogue of the expected codimension bounds for Andreotti–Mayer loci; the supplied text gives no resolution.