Permutation-module decomposition conjecture for Hessenberg-variety cohomology

Let $k$ be a positive integer, let $h\colon [k]\rightarrow [k]$ be a Hessenberg function, let $S$ be a regular semisimple operator on $\mathbb C^k$, and let $\mathfrak S_k$ act on $H^{2d}(\operatorname{Hess}(S,h))$ by Tymoczko's dot action. For a partition $\lambda=(\lambda_1,\ldots,\lambda_\ell)$ of $k$, let $\mathfrak S_\lambda$ be the corresponding Young subgroup and define the permutation module $M^\lambda=\operatorname{Ind}_{\mathfrak S_\lambda}^{\mathfrak S_k}{\bf 1}$. Permutation-module decomposition conjecture. For every $d$, the $\mathbb C\mathfrak S_k$-module $H^{2d}(\operatorname{Hess}(S,h))$ is a direct sum of permutation modules $M^\lambda$. This is the representation-theoretic reformulation of the refined Stanley–Stembridge conjecture; its general validity remains open.