Haefliger–Thurston homological conjecture for diffeomorphism groups

Let $M$ be an $n$-manifold, and let $\mathrm{BDiff}_c^{r}(M)^{\delta}$ denote the classifying space of the discrete group of compactly supported $C^r$ diffeomorphisms, while $\mathrm{BDiff}^r_c(M)$ denotes the classifying space of the corresponding topological group. Haefliger–Thurston's conjecture. The map

$$\eta\colon \mathrm{BDiff}_c^{r}(M)^{\delta}\to \mathrm{BDiff}^r_c(M)$$

induces an isomorphism on homology in degrees less than $n+1$ and a surjection on homology in degree $n+1$. By the Mather–Thurston theorem, this is equivalent to the associated bundle-theoretic formulation of the Haefliger–Thurston conjecture; the source gives no resolution status.