BBASV's invariant-center isomorphism conjecture for the small quantum group

Let $G$ be a complex simple simply connected algebraic group, let $\fu_\zeta^\vee$ be the small quantum group associated to the Langlands dual Lie algebra at a primitive $\ell$-th root of unity, and let $\Gr^{\zeta,\gamma}$ be the corresponding affine Springer fiber with extended affine Weyl group $\widetilde{W}$ acting on its cohomology. The existing construction gives an algebra embedding

$$H^*(\Gr^{\zeta,\gamma})^{\widetilde{W}}\subseteq Z(\fu_\zeta^\vee)^{G^\vee}.$$

BBASV's conjecture. The embedding above is an isomorphism.

This predicts that the geometrically constructed subalgebra exhausts the $G^\vee$-invariant part of the center of the small quantum group. The source presents the claim as conjectural and gives no resolution.