Generalized Shareshian–Wachs conjecture for proper circular arc digraphs

Let $\overrightarrow{G}$ be a proper circular arc digraph, meaning a digraph with no induced subdigraph isomorphic to either $\overrightarrow{K_{12}}$ or $\overrightarrow{K_{21}}$. Let

$$X_{\overrightarrow{G}}({\bf x},t)=\sum_{j=0}^{m}a_j({\bf x})t^j$$

be its chromatic quasisymmetric function, and call this palindromic polynomial $e$-unimodal when $a_{j+1}({\bf x})-a_j({\bf x})$ is $e$-positive for $0\leq j<\frac{m-1}{2}$. Generalized Shareshian–Wachs conjecture. The polynomial $X_{\overrightarrow{G}}({\bf x},t)$ is $e$-positive and $e$-unimodal. This generalizes the Shareshian–Wachs conjecture from unit interval digraphs to proper circular arc digraphs; the source does not specify its resolution status.