The $b[200~\Lambda\Sigma=\Delta$ character conjecture for cluster parking functions and diagonal coinvariantsb[201~

Let $A=\mathbb{C}[X_1,\dots,X_n,Y_1,\dots,Y_n]$ be the polynomial algebra with diagonal action of $\mathfrak{S}_n$, let $I\subset A$ be the ideal generated by constant-term-free $\mathfrak{S}_n$-invariant polynomials, and let $A/I$ be the diagonal coinvariants. The group $\operatorname{Dih}_{n+2}\times\mathfrak{S}_n$ acts on both $A/I$ and the reduced homology group $\tilde H_{n-2}(\operatorname{CPF}_n)$. Let $\Lambda$ be the character on the latter, $\Sigma$ a sign character of $\operatorname{Dih}_{n+2}$ extended to the product, and $\Delta$ the character on $A/I$.

The $\Lambda\Sigma=\Delta$ conjecture. We have $\Lambda\Sigma=\Delta$.

This conjecture proposes an equality between the dihedral-symmetric-group representation on cluster parking-function homology and diagonal coinvariants, extending dihedral sieving phenomena for $q,t$-Catalan objects. Its status is not specified in the supplied text.