The center–diagonal coinvariant algebra conjecture for small quantum $\mathfrak{sl}_m$

Let $u_q(\mathfrak{sl}_m)$ be the small quantum group, and let $\mathsf z_0:=\mathsf z_0(\mathfrak{sl}_m)$ denote the center of its principal block. Let $\mathrm{DC}_m$ be Haiman's diagonal coinvariant algebra for $S_m$, equipped with its standard bigrading, and let $\mathrm{C}_m\subset\mathsf z_0$ be the coinvariant subalgebra. The flag variety is $X=SL_m(\mathbb C)/B$, with complex dimension ${m\choose 2}$. For $i,j\in\mathbb N$, write

$$\mathsf z_0^{i,j}=\mathrm{H}^{i}(\widetilde{\mathcal N},\wedge^jT\widetilde{\mathcal N})^{-i-j}.$$

The center–diagonal coinvariant algebra conjecture. In type $A$, there exists an $S_m$-action on $\mathsf z_0$ extending the action on $\mathrm{C}_m$, commuting with the $\mathfrak{sl}_2$-action, such that, as a bigraded $S_m$-representation,

$$\mathsf z_0\cong \mathrm{DC}_m\otimes\operatorname{sgn},$$

where $\operatorname{sgn}$ is the one-dimensional sign representation in bidegree $(0,0)$, and

$$\mathsf z_0^{i,j}\cong\left(\mathrm{DC}_m\otimes\operatorname{sgn}\right)^{{m\choose 2}-\frac{i+j}{2},\frac{j-i}{2}}.$$

In particular,

$$\dim\mathsf z_0(\mathfrak{sl}_m)=\dim\mathrm{DC}_m=(m+1)^{m-1}.$$

This conjecture extends the computed cases in type $A$ and would give a precise representation-theoretic and bigraded description of the principal-block center for all $m$.