e-positivity and e-unimodality conjecture for circular indifference digraphs

Let $\overrightarrow{G}=(V,E)$ be a circular indifference digraph. Write its chromatic quasisymmetric function as

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

For a symmetric function, $e$-positive means that its expansion in the elementary symmetric-function basis has nonnegative coefficients.

Circular indifference digraph conjecture. The palindromic polynomial $X_{\overrightarrow{G}}({\bf x},t)$ is $e$-positive and $e$-unimodal: $a_j({\bf x})$ is $e$-positive for every $j$, and $a_{j+1}({\bf x})-a_j({\bf x})$ is $e$-positive for every $j\leq\frac{|E|-1}{2}$.

The conjecture generalizes the Shareshian–Wachs $e$-positivity conjecture from natural unit interval graphs to circular indifference digraphs. The paper proves the claim for directed cycles and presents that result as evidence; the general status is not resolved in the supplied text.