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:



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> 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?
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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.



Finitistic dimension via reflexive modules
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.


