Babson–Kozlov's contractibility conjecture for rank-selected partition-lattice quotients

Let $\Pi_n$ be the partition lattice, let $S$ be a set of ranks, and let $\Pi_n^S$ denote its rank-selected subposet. Write $\Delta(\Pi_n^S)$ for the order complex of $\Pi_n^S$, let $S_n$ act by permuting the underlying $n$ elements, and let $b_S(n)$ be the number of facets with minimal new face colored by $S$ in the partitioning of $\Delta(\Pi_n)/S_n$. Babson–Kozlov's conjecture. The rank-selected quotient complex $\Delta(\Pi_n^S)/S_n$ is contractible if and only if $b_S(n)=0$. This conjecture relates the combinatorial partitioning statistic $b_S(n)$ to the topology of the quotient complex; the supplied source identifies it as Conjecture 4.11 of Babson and Kozlov, but provides no resolution evidence.