Haefliger–Thurston connectivity conjecture for trivial-normal Haefliger structures

Recall that $\overline{\mathrm{B}\Gamma}^r_k$ classifies codimension $k$ $C^r$-Haefliger structures with trivial normal bundle. Haefliger–Thurston's conjecture. The space $\overline{\mathrm{B}\Gamma}^r_k$ is $2k$-connected, that is, $\pi_i(\overline{\mathrm{B}\Gamma}^r_k)=0$ for $i\leq 2k$. This is a strengthening of Haefliger's known connectivity result and is related, via the Mather–Thurston theorem, to homological stability for diffeomorphism groups; its general status is not specified in the source.