Auslander's conjecture for semisimple Hopf actions on AS regular algebras

Let $A$ be a noetherian AS regular algebra, and let $H$ be a semisimple Hopf algebra acting on $A$ under the standing hypotheses, with trivial homological determinant.

Auslander's conjecture. The skew group ring $A\#G$ is naturally isomorphic to

$$\operatorname{Hom}_{(A^G)^{op}}(A,A)$$

as graded algebras.

This conjecture extends Auslander's theorem from finite group actions on polynomial rings to semisimple Hopf algebra actions on noetherian AS regular algebras. The statement is known in dimension two under the stated hypotheses, while it is open in arbitrary dimension.