Recall that a module $M$ over a ring $R$ is reflexive in case the natural evaluation map $f_M:M \rightarrow M^{**}$ (where $M^{*}=Hom_R(M,R)$) is an isomorphism, where $f_M(m)=g$ with $g(h)=h(m)$, when $h \in Hom_R(M,R)$. For example every projective module is reflexive. Assume that all modules in the following are finitely generated. Call a reflexive module non-trivial in case it is not projective.

Question: Does an arbitrary ring of finite global dimension have global dimension at most two if and only if there is no non-trivial reflexive module?

Here two similar questions restricted to finite dimensional algebras:

Does a finite dimensional algebra have a non-trivial reflexive module in case it has finitistic dimension at least 3?

Does a finite dimensional algebra of finite global dimension have global dimension at most two if and only if it has no non-trivial reflexive modules?

In general a non-trivial reflexive module implies that the global dimension of a ring with finite global dimension is at least three but I do not know about the other direction. The answer should be positive for QF-3 algebras.