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If $R$ is a Gorenstein domain of dimension n = 2,3,4, then finite reflexive modules are projective.
Proof: Since both are local properties, we assume $(R,m)$ is local. As Greg pointed out, $\operatorname{Ext}_R^1(M,R) = \operatorname{Ext}^2_R(M,R) = 0$. By local duality, this says $H^{n-1}_m(M) = H^{n-2}_m(M) = 0$. We need to show that $\operatorname{Ext}^i_R(M,R)$ vanish for all $i>2$, iff the lower local cohomologies $H^{n-i}_m(M)$ vanish. But this is clear since $R$ has depth $\ge 2$, any $\operatorname{Hom}_R(N,R)$ will have depth $\ge 2$ (see https://stacks.math.columbia.edu/tag/0AV5), in particular $M$ being reflexive has depth $\ge 2$.