Stable cohomology isomorphism conjecture for relative completion

For a partition ${\boldsymbol\mu}$ with at most $g$ parts, let $V_{\boldsymbol\mu}$ be the corresponding irreducible ${\mathrm{Sp}}(H)$-module, and let

$$H^{\bullet}({\mathcal G}_{g,n+\vec r};V_{\boldsymbol\mu}) \to H^{\bullet}({\Gamma}_{g,n+\vec r};V_{\boldsymbol\mu})$$

be the homomorphism induced by the map from the mapping class group to its relative completion. Stable cohomology isomorphism conjecture. The homomorphism with $V=V_{\boldsymbol\mu}$ is stably an isomorphism for all ${\boldsymbol\mu}$. This is known in genera $0$ and $1$, while the comparison fails to be an isomorphism in all degrees for every $g\geq 3$; the stable assertion remains open.