Finitistic dimension via reflexive modules

Recall that an $A$-module $M$ is reflexive in case the natural evaluation map $M \rightarrow M^{**}$ is an isomorphism, where $M^{*}=Hom_A(M,A)$.

Question:

Given a finite dimensional algebra $A$ of finitistic dimension at least two, is the finitistic dimension of $A$ equal to the supremum of finite projective dimensions of reflexive modules+2?

It is true for QF-3 algebras, which include for example Nakayama algebras. But I would think it fails in general but I do not know a counterexample.