Stable finite-dimensionality of Torelli and IA_n homology

Let $\mathcal{I}_{n,1}$ be the Torelli group of a genus-$n$ surface with one boundary component and let $\operatorname{IA}_n$ be the IA-automorphism group of the free group $F_n$. For a group representation $V$, let $V^{\mathrm{fd}}$ be its finite-dimensional part. Stable finite-dimensionality conjecture. For each $i\geq 1$ and each $n$ sufficiently large depending on $i$, the inclusions

$$H_i(\mathcal{I}_{n,1};\mathbb{Q})^{\mathrm{fd}}\hookrightarrow H_i(\mathcal{I}_{n,1};\mathbb{Q})$$

and

$$H_i(\operatorname{IA}_n;\mathbb{Q})^{\mathrm{fd}}\hookrightarrow H_i(\operatorname{IA}_n;\mathbb{Q})$$

are isomorphisms. This conjecture predicts that, in the stable range, the relevant homology representations have no infinite-dimensional part; the source gives no resolution.