Haefliger–Thurston conjecture on the connectivity of diffeomorphism classifying spaces

Let $M$ be an oriented closed manifold, and let $\mathrm{Diff}_c^r(M)$ be the group of compactly supported, orientation-preserving $C^r$ diffeomorphisms with the $C^r$-Whitney topology. Write $\mathrm{Diff}_c^r(M)^\delta$ for the same group with the discrete topology. The identity homomorphism induces

$$\eta:\mathrm{B}\mathrm{Diff}_c^r(M)^\delta\to\mathrm{B}\mathrm{Diff}_c^r(M).$$

Haefliger–Thurston conjecture. The map $\eta$ is a homology isomorphism in degrees at most $\operatorname{dim}(M)$ and is surjective on homology in degree $\operatorname{dim}(M)+1$.

This is the classifying-space formulation of the Haefliger–Thurston conjecture and is related by Mather–Thurston theory to the flatness statement for manifold bundles. The source gives no evidence that the conjecture has been resolved.