Ellzey's e-positivity conjecture for proper circular arc digraphs

Let $\vec G$ be a proper circular arc digraph: its vertices correspond to arcs on a circle whose left endpoints occur in clockwise order and which do not contain one another, with $(i,j)$ a directed edge exactly when the left endpoint of the $j$th arc lies in the $i$th arc. Let $X_{\vec G}(\bm x;q)$ be its chromatic quasisymmetric function, weighting proper colourings by $q^{\operatorname{asc}(\kappa)}$. The source states that this function is symmetric for such digraphs. It is $e$-positive if its expansion in the elementary basis has coefficients that are polynomials in $q$ with nonnegative coefficients.

Ellzey's conjecture. If $\vec G$ is a proper circular arc digraph, then $X_{\vec G}(\bm x;q)$ is $e$-positive.

This extends the refined Stanley–Stembridge conjecture from natural unit interval graphs to proper circular arc digraphs. The source states that this extension is open.