Finite generation of the ring of holomorphic functions with polynomial growth on the Kähler-Ricci shrinker

Let (X, g, J, f ) be a non-compact gradient shrinking Kahler-Ricci soliton. We prove that if the scalar curvature of X satisfies a mild assumption, then OP (X), the ring of holomorphic functions with polynomial growth on X, is finitely generated. This gives a partial confirmation to a conjecture of Munteanu and Wang (cf.[MW14]).