Shareshian–Wachs q-analogue of the Stanley–Stembridge conjecture

Let $G$ be the incomparability graph of a natural unit interval order, that is, a poset that is both $(\mathbf{3}+\mathbf{1})$-free and $(\mathbf{2}+\mathbf{2})$-free. Let $X_G(\mathbf{x},q)$ denote the Shareshian–Wachs chromatic quasisymmetric function, which is symmetric for such $G$. A symmetric function in $\mathbf{x}$ with coefficients in $\mathbb{N}[q]$ is $e$-positive if its expansion in the elementary symmetric-function basis has all coefficients in $\mathbb{N}[q]$.

Shareshian–Wachs conjecture. If $G$ is the incomparability graph of a natural unit interval order, then $X_G(\mathbf{x},q)$ is $e$-positive.

This is a $q$-analogue of the Stanley–Stembridge conjecture, with the parameter $q$ recording ascents in proper colorings. It refines the corresponding positivity problem for natural unit interval orders; the source provides no resolution of this assertion.