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More to the point: if $A$ is a left noetherian ring of global dimension at most $2$, then every finitely generated reflexive left $A$-module is projective. Indeed, if $M$ is a finitely generated module and $$P_1\to P_0\to M\to 0$$ is a projective presentation by finitely generated projectives, applying the functor $(\mathord-)^*=\hom_A(\mathord-,A)$ we get an exact sequence $$0\to M^*\to P_0^*\to P_1^*\to E\to 0,$$ when $E$ is just the cokernel of the map $P_0^*\to P_1^*$. Since $\mathrm{pdim}\,E\leq 2$ and since $P_0^*$ and $P_1^*$ are projective, the kernel of $P_0^*\to P_1^*$, namely $M^*$, must be projective. This shows that the dual of finitely generated module is projective, so a finitely generated reflexive, being isomorphic to the dual of its dual module, is projective.