The zeta-extension conjecture for compactly supported cohomology of the moduli of abelian varieties

For each odd integer $g\geq 3$, let $\mathcal{A}_g$ be the moduli space of principally polarized abelian varieties of dimension $g$, and let $H^{2g}_c(\mathcal{A}_g)$ carry its mixed Hodge structure. A subquotient isomorphic to an extension of $\mathbb{Q}(-g)$ by $\mathbb{Q}$ is an exact sequence of mixed Hodge structures

$$0\longrightarrow\mathbb{Q}(-g)\longrightarrow E\longrightarrow\mathbb{Q}\longrightarrow 0.$$

Zeta-extension conjecture. The mixed Hodge structure on $H^{2g}_c(\mathcal{A}_g)$ has a subquotient isomorphic to such an extension whose class is given by a nonzero rational multiple of $\zeta(g)$. This predicts a higher-genus generalization of the established nontrivial $\zeta(3)$ extension in genus $3$.