Refined Stanley–Stembridge conjecture for Hessenberg graphs

Let $k$ be a positive integer, let $h\colon [k]\rightarrow [k]$ be a Hessenberg function, and let $G_h$ be the graph associated with $h$. For a proper coloring $\kappa$ of $G_h$, write $\operatorname{asc}(\kappa)$ for the number of edges oriented from a smaller to a larger vertex whose endpoint colors increase, and let $X_{G_h}({\bf x};q)$ be the chromatic quasisymmetric function. Let $e_\lambda({\bf x})$ denote the elementary symmetric function indexed by a partition $\lambda$ of $k$. Refined Stanley–Stembridge conjecture. The function $X_{G_h}({\bf x};q)$ is $e$-positive:

$$X_{G_h}({\bf x}; q)=\sum_{\lambda\vdash k} c_\lambda(q)e_\lambda({\bf x}),$$

where $c_\lambda(q)$ is a polynomial in $q$ with nonnegative coefficients for every partition $\lambda$ of $k$. The ordinary Stanley–Stembridge conjecture is now known, but this chromatic-quasisymmetric refinement remains open.