On the category $\mathcal{O}$ for generalized Weyl algebras

Let $H(R, \phi, z)$ be a generalized Weyl algebra associated with a ring $R$, its central element $z\in Z(R)$ and an automorphism $\phi,$ such that for some $l \geq 1$, $\phi^l(z)-z$ is nilpotent and $(z,\phi^i(z))=R$ for all $0<i<l$. We prove that the category $\mathcal{O}$ over $H(R, z,\phi)$ is equivalent to the category $\mathcal{O}$ over its $l$-th twist the generalized Weyl algebra $H(R, z,\phi^l).$ This result is significantly more general than the corresponding one for the Weyl algebra over $\mathbb{Z}/p^n\mathbb{Z}.$