Cofiniteness of local cohomology modules and subcategories of modules

Let R be a commutative noetherian ring and I an ideal of R. Assume that for all integers i the local cohomology module \({\text {H}}_I^i(R)\) is I-cofinite. Suppose that \(R_\mathfrak {p}\) is a regular local ring for all prime ideals \(\mathfrak {p}\) that do not contain I. In this paper, we prove that if the I-cofinite modules form an abelian category, then for all finitely generated R-modules M and all integers i, the local cohomology module \({\text {H}}_I^i(M)\) is I-cofinite.