Extension of the Auslander theorem for noetherian graded down-up algebras

Let $R=A(\alpha,\beta)$ be a noetherian graded down-up algebra generated by $V:=\Bbbk x+\Bbbk y$, and let $G$ be a nontrivial finite subgroup of $\operatorname{Aut}_{gr}(R)$. Assume $\operatorname{char}\,\Bbbk=0$. The theorem asserts the conclusions stated for the cases either $\beta\neq -1$ or $(\alpha,\beta)=(2,-1)$: $\sf p(R,G)\geq 2$, the $G$-action on $R$ is homologically small, and there is a natural isomorphism of graded algebras

$$R\ast G\cong\operatorname{End}_{R^G}(R).$$

Extension conjecture. Theorem ${\rm{}}$ also holds when $\beta=-1$ and $\alpha\neq 2$. This would extend the noncommutative Auslander theorem to the remaining parameter cases for noetherian graded down-up algebras covered by this condition.