The noncommutative Shephard–Todd–Chevalley conjecture for AS regular algebras

Let $A$ be an AS regular algebra, and let $G$ be a finite group of graded automorphisms of $A$. A reflection of $A$ is an automorphism of the kind defined in the surrounding discussion, including classical and mystic reflections. Noncommutative Shephard–Todd–Chevalley conjecture. The invariant subring $A^G$ is AS regular if and only if $G$ is generated by reflections of $A$. This conjecture proposes the invariant-theoretic characterization of AS regular fixed subrings that generalizes the classical Shephard–Todd–Chevalley theorem; the survey does not provide a resolution in general.