An analogue of a theorem due to Levin and Vasconcelos

Let $(R,\m)$ be a Noetherian local ring. Consider the notion of homological dimension of a module, denoted H-dim, for H= Reg, CI, CI$_*$, G, G$^*$ or CM. We prove that, if for a finite $R$-module $M$ of positive depth, $\Hd_R({\m}^iM)$ is finite for some $i \geq \reg(M)$, then the ring $R$ has property H.