Contact Haefliger–Thurston conjecture

Let $(M,\alpha)$ be a contact manifold of dimension $n=2k+1$, where $\alpha$ is a smooth $1$-form such that $\alpha\wedge(d\alpha)^n$ is a volume form. Let $\mathrm{Cont}_c(M,\alpha)$ be the group of compactly supported, orientation-preserving smooth contactomorphisms, namely smooth diffeomorphisms $f$ satisfying $f^*\alpha=\lambda_f\alpha$ for a positive smooth function $\lambda_f$. Give $\mathrm{Cont}_c(M,\alpha)^\delta$ the discrete topology. The natural map

$$\mathrm{B}\mathrm{Cont}_c(M,\alpha)^\delta\to\mathrm{B}\mathrm{Cont}_c(M,\alpha)$$

Contact Haefliger–Thurston conjecture. The natural map induces a homology isomorphism through degree $n$ and a surjection on homology in degree $n+1$.

This is the analogue of the Haefliger–Thurston conjecture for foliations with transverse contact structure. The source presents it as a formulation motivated by the vanishing range of continuous Lie algebra cohomology for formal contact vector fields, and gives no resolution.