Sharp stabilization conjecture for rank-selected homology of the partition lattice

Let $\Pi_n$ be the partition lattice, let $S\subset\{1,2,\ldots,n-2\}$, and set $i=\max(S)$. Let $\beta_S(\Pi_n)$ denote the rank-selected homology $S_n$-representation. Sharp stabilization conjecture. The representation $\beta_S(\Pi_n)$ stabilizes sharply at

$$n=4i-(|S|-1).$$

The conjecture is motivated by preliminary analysis of $\beta_S(\Pi_n)$, and the supplied text gives no evidence that the claimed sharp threshold has been proved or disproved.