The Madsen-Weiss Theorem, as described by Hatcher, states that there is an isomorphism $H_*( \mathscr{C}_{\infty})\cong H_*(\Omega_0^{\infty}AG^+_{\infty,2})$ where $\Omega_0^{\infty}AG^+_{\infty,2}$ denotes the base point path-connected component of $\Omega^{\infty}AG^+_{\infty,2}$ for the direct limit $\mathscr{C}_{\infty}=\cup_g\mathscr{C}_g$ of $\mathscr{C}_g$, the space of subsurfaces of $(-\infty,g]\times \mathbb{R}^{\infty}$ diffeomorphic to $S_{g,1}$ (the compact surfaces of genus $g$ with one boundary component).

However, is there an isomorphism which generalizes past $d=2$ using the scanning map $\mathscr{C}(M,\mathbb{R}^{\infty})\to\Omega^{\infty}AG^+_{\infty,d}$? That is, is there an isomorphism involving $H_*(\mathscr{C}(M,\mathbb{R}^{\infty}))$ and $H^*(\Omega_0^{\infty}AG^+_{\infty,d})$?

Note $\mathscr{C}(M,\mathbb{R}^{\infty})$ denotes the space of all smooth oriented submanifolds of $\mathbb{R}^{\infty}$ diffeomorphic to $M$ and $\Omega^{\infty}AG^+_{\infty,d}$ is the limit of the $n$-fold loop space of the one-point compactification $AG^+_{n,d}$ of $AG_{n,d}$, the affine Grassmannian of oriented flat $d$-planes in $\mathbb{R}^n$.

I came across this paper, which I believe is the appropriate generalization if we replace the manifold $W_{g,1}=\#^g(S^n\times S^n)-int(D^{2n})$ with $W_{g,1}=\#^gS^n-int(D^n)$, although I am not certain.

See Hatcher's paper for better reference. Thanks in advance!