  We show that the small and large restricted injective dimensions coincide for
Cohen-Macaulay rings of finite Krull dimension. Based on this, and inspired by
the recent work of Sather-Wagstaff and Totushek, we suggest a new definition of
Cohen-Macaulay Hom injective dimension. We show that the class of
Cohen-Macaulay Hom injective modules is the right constituent of a perfect
cotorsion pair. Our approach relies on tilting theory, and in particular, on
the explicit construction of the tilting module inducing the minimal tilting
class recently obtained in arXiv:2207.01309.
