$\DeclareMathOperator{\Diff}{Diff}$ From the work of Galatius - Randall-Williams and Berglund - Madsen we have homological stability (with respect to g) of $B\Diff_\partial (W_{g,1})$ and rational homological stability of $B\widetilde\Diff_\partial(W_{g,1})$, where $W_{g,1}$ is $(\#_g S^d \times S^d ) \setminus D^{2d}$ and $\widetilde\Diff$ denotes block diffeomorphisms (essentially an analogue of diffeomorphisms where path components are naturally pseudoisotopy classes).

By analyzing the Serre spectral sequence for the fibration $\widetilde\Diff_\partial(W_{g,1}) / \Diff_\partial (W_{g,1}) \rightarrow B\Diff_\partial (W_{g,1}) \rightarrow B\widetilde\Diff_\partial(W_{g,1}) $ we deduce from the aforementioned homological stability results, that $H_*(\widetilde\Diff_\partial(W_{g,1}) / \Diff_\partial (W_{g,1});\mathbb Q)_{\pi_0(\widetilde\Diff_\partial(W_{g,1}))} $ has stability with respect to $g$. By pseudo-isotopy implies isotopy and some of Wall's work on highly connected manifolds, one could replace $\pi_0(\widetilde\Diff_\partial(W_{g,1}))$ by a certain arithmetic group $\Gamma$ if she wished.

It is known that $\widetilde\Diff_\partial(W_{g,1}) / \Diff_\partial (W_{g,1})$ is related to Waldhausen A-theory, see Weiss' and Williams' Automorphisms of manifolds and algebraic K-theory: I. Is there a known proof using A-theory that the coinvariants of these homology groups have stability?