Unimodality and palindromicity conjecture for the flag descent polynomial

Let $\mathcal{F}_n^\alpha(x)$ be the flag descent polynomial associated with the quotient $\mathbb{Z}_{\alpha}\wr \mathfrak{S}_n/H$. Unimodality and palindromicity conjecture. The polynomial $\mathcal{F}_n^\alpha(x)$ is unimodal and palindromic. This is motivated by computational evidence, alongside the paper's observation of a symmetric distribution of the flag descent statistic over the quotient; the conjectured unimodality remains unresolved in the supplied source.