Shareshian–Wachs Hessenberg variety representation conjecture

Let ${\bf m}$ be a Hessenberg vector, and let $P({\bf m})$ be the natural unit interval order on $[n]$ defined by $i<_{P({\bf m})}j$ if $j>m_i$. Let ${\mathcal H}({\bf m})$ be the associated Hessenberg variety, let $X_{{\rm inc}(P({\bf m}))}({\mathbf x},t)$ be the chromatic quasisymmetric function of its incomparability graph, let $\omega$ be the involution on symmetric functions, and let ${\rm ch}$ denote the Frobenius characteristic. Shareshian and Wachs's conjecture. For all Hessenberg vectors ${\bf m}$,

$$\omega X_{{\rm inc}(P({\bf m}))}({\mathbf x},t)=\sum_{j\geq 0}{\rm ch}\bigl(H^{2j}({\mathcal H}({\bf m}))\bigr)t^j.$$

This connects chromatic quasisymmetric functions with the symmetric-group representations on the cohomology of Hessenberg varieties. The source notes that the conjecture holds for ${\bf m}=(2,3,\ldots,n)$, while the general case remains open, including questions about faithful symmetric-group actions.