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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.

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My answer to your other question https://mathoverflow.net/a/319407/22989 gives an example of a finite dimensional algebra with global dimension three and no non-projective reflexive modules, which also gives a negative answer to this question.

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