A stable splitting of factorisation homology of generalised surfaces

For a manifold $W$ and an $E_d$-algebra $A$, the factorisation homology ${\int }_{W}A$ can be seen as a generalisation of the classical configuration space of labelled particles in $W$. It carries an action by the diffeomorphism group ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_{\mathrm{\partial }}(W)$, and for the generalised surfaces $W_{g,1}:=({\mathrm{\#}}^g{S}^n\times S^n)\setminus \mathring{D}{}^{2n}$, we have stabilisation maps among the quotients ${\int }_{W_{g,1}}A\mathrm{/}\text{\hspace{0.17em}⁣}\mathrm{/}{\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_{\mathrm{\partial }}(W_{g,1})$ which increase the genus $g$. In the case where a highly-connected tangential structure $\theta$ is taken into account, we describe its stable homology in terms of the iterated bar construction ${\mathrm{B}}^{2n}A$ and a tangential Thom spectrum $\mathrm{M}\mathrm{T}\theta$. We also consider the question of homological stability.