The folklore conjecture on stable Torelli-group homology

Let $\operatorname{\mathcal{I}}_g$ be the Torelli group of a closed genus-$g$ surface, and let $\operatorname{Sp}_{2g}(\mathbb{Z})$ act on its rational homology through the quotient of the mapping class group. For each integer $k$, consider the homology group

$$\operatorname{H}_k(\operatorname{\mathcal{I}}_g;\mathbb{Q}).$$

Folklore conjecture. For each $k$, there exists some $G_k$ such that for $g \geq G_k$, the homology group $\operatorname{H}_k(\operatorname{\mathcal{I}}_g;\mathbb{Q})$ is finite-dimensional and the action of $\operatorname{Sp}_{2g}(\mathbb{Z})$ on it extends to a rational representation of the algebraic group $\operatorname{Sp}_{2g}(\mathbb{Q})$. Johnson's work shows that this holds for $k=1$ with $G_1=3$, but it remains open for all $k\geq 2$.