Babson–Kozlov connectivity conjecture for Hom complexes

Let $G$ be a graph with maximal degree $d$, and let $K_n$ be the complete graph on $n$ vertices. Babson–Kozlov's conjecture. The complex $\operatorname{Hom}(G,K_n)$ is at least $(n-d-2)$-connected. This conjecture was posed by Babson and Kozlov and is stated in the paper as a conjecture; the supplied material gives no resolution status.