Shareshian–Wachs conjecture for the matching complex homology representation

Let $n=2k$ be even. The complexes ${\mathsf {NPM}}_n$ and the poset ${\mathcal L}_n^{(2)}$ have reduced homology groups carrying actions of $S_n$, where $\varepsilon_n$ denotes the sign character of $S_n$. The poset ${\mathcal L}_n^{(2)}$ consists of trees with $n$ bijectively labelled leaves whose nonleaves have degree congruent to $2$ modulo $2$ but are not of degree $2$, and its order-complex homology is concentrated in dimension $t$. Shareshian–Wachs conjecture. There is an isomorphism of $S_n$-modules

$$\widetilde{H}_{3k-4}({\mathsf {NPM}}_n) \cong_{S_n} \widetilde{H}_t(\Delta {\mathcal L}_n^{(2)}) \otimes \varepsilon_n.$$

This strengthens the stated Shareshian–Wachs theorem, which identifies the restriction of the left-hand side to $S_{n-1}$ with the corresponding homology representation. The conjecture concerns the full $S_n$-module structure and is unresolved in the supplied source.