The noncommutative Auslander conjecture for semisimple Hopf actions

Let $\operatorname{\Bbbk}$ be an algebraically closed field of characteristic zero. Let $R$ be a noetherian connected graded Artin–Schelter regular algebra, and let $H$ be a semisimple Hopf algebra acting homogeneously and inner faithfully on $R$. Write $R^H$ for the invariant subring, and let the homological determinant of the $H$-action be trivial. The Auslander map is

$$\varphi:R\#H\longrightarrow \operatorname{End}_{R^H}(R),\qquad r\#h\longmapsto\bigl(x\longmapsto r(h\operatorname{\rightharpoonup}x)\bigr).$$

Noncommutative Auslander conjecture. The map $\varphi$ is an isomorphism of graded algebras.

This conjecture asks for a noncommutative analogue of Auslander's theorem for finite linear groups and is connected with the McKay correspondence. It was stated as an open question in the cited work; the supplied source does not indicate a resolution.