The stable algebraic Albanese cohomology conjecture for Torelli groups

Let $\mathcal{I}_g$, $\mathcal{I}_{g,1}$ and $\mathcal{I}_g^1$ be the Torelli groups of a closed surface, a surface with one boundary component, and a surface with one marked point, respectively. Let $X”_*$, $Y”_*$ and $Z”_*$ be the graded $\operatorname{Sp}(2g,\mathbb{Q})$-representations defined in the source, and let $\widetilde{S}^*$ denote the traceless graded-symmetric algebra. The Torelli Albanese cohomology conjecture. Stably, there are graded $\operatorname{Sp}(2g,\mathbb{Q})$-isomorphisms

$$H_A^*(\mathcal{I}_g,\mathbb{Q})\cong\widetilde{S}^*(X”_*),\qquad H_A^*(\mathcal{I}_{g,1},\mathbb{Q})\cong\widetilde{S}^*(Y”_*),\qquad H_A^*(\mathcal{I}_g^1,\mathbb{Q})\cong\widetilde{S}^*(Z”_*).$$

The source presents these as conjectural stable structures; the claim remains open in general.