The double coinvariant algebra conjecture for the center of the small quantum group of type A

Let $\mathfrak{u}_0(\mathfrak{sl}_m)$ be the small quantum group at the relevant root of unity, let $\mathbf{z}(\mathfrak{u}_0(\mathfrak{sl}_m))$ denote its center, and let $\mathrm{DC}_m$ be the double coinvariant algebra. The double coinvariant algebra conjecture. As a bigraded vector space, there is an isomorphism

$$\mathbf{z}(\mathfrak{u}_0(\mathfrak{sl}_m))\cong\mathrm{DC}_m.$$

In particular,

$$\dim \mathbf{z}(\mathfrak{u}_0(\mathfrak{sl}_m))=(m+1)^{m-1}.$$

This conjecture is motivated by computations for $\mathfrak{sl}_3$; its general validity is not resolved in the supplied text.