The mystic reflection group conjecture for quantum polynomial rings

Let $B$ be a quantum polynomial ring and let $G$ be a finite subgroup of $\operatorname{Aut_{gr}}(B)$ such that the fixed subring $B^G$ has finite global dimension. For a positive integer $n$ and parameters $\alpha$ and $\beta$, let $M(n,\alpha,\beta)$ denote the mystic reflection group appearing in the paper. Mystic reflection group conjecture. The group $G$ is a product of classical reflection groups and copies of $M(n,\alpha,\beta)$. This conjecture refines the preceding invariant-theoretic conjecture by predicting the structure of every finite group whose fixed subring has finite global dimension; the paper establishes the corresponding structural statement under the skew-polynomial hypotheses, while the quantum-polynomial-ring generalization remains open.