Circular chromatic coefficient palindromicity and unimodality conjecture

Let $\mathbf{a}$ be a circular area sequence, and write the corresponding chromatic symmetric function as

$$\mathrm{X}_{\mathbf{a}}(\mathbf{x};q)=\sum_\lambda c_\lambda(q)\mathrm{e}_\lambda(\mathbf{x}).$$

Circular chromatic coefficient conjecture. The coefficients $c_\lambda(q)$ are palindromic and unimodal polynomials with non-negative integer coefficients. This extends and refines the circular e-positivity conjecture and is supported by the special cases proved in the paper; it remains open.