The invariant-center conjecture for semisimple small quantum groups

Let $\mathfrak{g}$ be a semisimple Lie algebra with Weyl group $W$, let $\mathfrak{u}_0(\mathfrak{g})$ be the corresponding small quantum group, let $\mathbf{z}(\mathfrak{u}_0(\mathfrak{g}))$ be its center, and let $\mathrm{DC}(W)$ be the double coinvariant algebra associated with $W$. The invariant-center conjecture.

$$\mathbf{z}(\mathfrak{u}_0(\mathfrak{g}))^{\mathfrak{g}}\cong\mathrm{DC}(W).$$

The conjecture is motivated by computations in type $\mathfrak{b}_2$; its general validity is not resolved in the supplied text.