Hersh–Reiner sharp stability conjecture for rank-selected homology of partition lattices

Let $\Pi_n$ be the poset of set partitions of $\{1,2,\dots,n\}$ ordered by refinement, and let $S\subseteq\{1,2,\dots,n-2\}$. Write $\beta_S(\Pi_n)$ for the rank-selected homology $\mathfrak{S}_n$-representation associated with $S$. Hersh–Reiner's sharp stability conjecture. The representation $\beta_S(\Pi_n)$ stabilizes sharply at $4\max S-|S|+1$. The conjecture concerns the precise onset of representation stability for rank-selected homology of partition lattices; the supplied source does not indicate whether it has been resolved.