Cho–Hong–Lee permutation-module generator conjecture for Hessenberg cohomology

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 $H^*(\operatorname{Hess}(S,h))$. For each $d$, let $\mathcal G_h^d$ be the distinguished set of permutations indexing the module generators $\sigma_{w,h}$ of $H^{2d}(\operatorname{Hess}(S,h))$, and let $\mathbb C\mathfrak S_k(\sigma_{w,h})$ denote the cyclic submodule generated by $\sigma_{w,h}$. Cho–Hong–Lee conjecture. For every $w\in\mathcal G_h^d$, there exists $\widehat{\sigma}_{w,h}\in\mathbb C\mathfrak S_k(\sigma_{w,h})$ such that $\mathbb C\mathfrak S_k(\widehat{\sigma}_{w,h})$ is a permutation module and

$$H^{2d}(\operatorname{Hess}(S,h))=\bigoplus_{w\in\mathcal G_h^d}\mathbb C\mathfrak S_k(\widehat{\sigma}_{w,h}).$$

This conjecture gives a basis-level refinement of the permutation-module decomposition conjecture and remains open in the stated generality.