The $q,t$-dihedral sieving conjecture for cluster parking functions

Let $\operatorname{Dih}_{n+2}$ act on the diagonal coinvariants $A/I$, and let $\varDelta$ denote their bigraded character as a character of $\mathfrak{S}_n$ polynomially depending on $q$ and $t$. Let $\Lambda$ be the character of $\operatorname{Dih}_{n+2}\times\mathfrak{S}_n$ on $\tilde H_{n-2}(\operatorname{CPF}_n)$, and let $\Sigma$ be the specified sign character of $\operatorname{Dih}_{n+2}$ extended to the product. For $\mathbf{F}\in\operatorname{Dih}_{n+2}$, write $\lambda_1,\lambda_2$ for the eigenvalues of the defining representation $\rho_1(\mathbf{F})\in O(\mathbb{R}^2)$.

The $q,t$-dihedral sieving conjecture. For every $\mathbf{F}\in\operatorname{Dih}_{n+2}$ and $\sigma\in\mathfrak{S}_n$,

$$(\Lambda\Sigma)(\mathbf{F},\sigma)=\left.\varDelta(\sigma)\right\vert_{q=\lambda_1,\ t=\lambda_2}.$$

This is the $q,t$-symmetric-function formulation of the preceding character identity and is intended as a homological dihedral sieving phenomenon. The supplied text says that its extension beyond symmetric groups is difficult because a sufficiently developed $q,t$-combinatorics is lacking.