Shareshian–Wachs conjecture for chromatic quasisymmetric functions

Let $G=(V,E)$ be a unit interval graph, with its vertices in the natural order. Define its chromatic quasisymmetric function by

$$\chi_G(x;q)=\sum_{\kappa}q^{\operatorname{asc}(\kappa)}x^\kappa,$$

where the sum is over proper colorings and $\operatorname{asc}(\kappa)$ counts edges $\{i,j\}$ with $i<j$ and $\kappa(i)<\kappa(j)$. Shareshian–Wachs conjecture. The function $\chi_G(x;q)$ is symmetric and $e$-positive. This is the quasisymmetric refinement of the Stanley–Stembridge conjecture. Although Hikita proved the Stanley–Stembridge conjecture, that proof does not imply the Shareshian–Wachs conjecture, which remains open.