Cohomological dimension conjecture for the moduli space of principally polarized abelian varieties

Let $\mathcal A_g$ denote the moduli space of principally polarized abelian varieties of dimension $g$, and let $\operatorname{cd}(\mathcal A_g)$ be its cohomological dimension. Cohomological dimension conjecture. In all characteristics,

$$\operatorname{cd}(\mathcal A_g)=\frac{g(g-1)}{2}.$$

The conjecture is motivated by the lower bound coming from the affine covering number and by complete subvarieties in characteristic $p$; the paper proves an upper bound of $6$ for $\mathcal A_4(\mathbb C)$ and a lower bound of $g(g-1)/2$ for the affine covering number. The equality in all characteristics remains open.