The permutation-basis conjecture for regular semisimple Hessenberg cohomology

Let $h:[n]\to[n]$ be a Hessenberg function, meaning a nondecreasing function with $h(i)\geq i$ for all $i$. Let $\mathcal{H}ess(\mathsf{S},h)$ be the regular semisimple Hessenberg variety associated with a regular semisimple $\mathsf{S}\in\mathfrak{gl}(n,\mathbb{C})$, and let the symmetric group $S_n$ act on its cohomology $H^*(\mathcal{H}ess(\mathsf{S},h))$ via the dot action. Permutation-basis conjecture. There exists a basis of $H^*(\mathcal{H}ess(\mathsf{S},h))$ that is permuted by the dot action, such that the stabilizer of each basis element is a reflection subgroup. The conjecture would provide a permutation-basis description of the dot-action representation and, according to the supplied context, would imply the Stanley–Stembridge conjecture. The supplied text gives no resolution status.