I am reading "Ample divisors on fine moduli spaces on the Projective plane" by Stromme. In the proof of Proposition 2.4, he seems to claim that if $E$ is a torsion free sheaf on a smooth variety and the singular locus of $E$ has codimension at least $3$, then $E$ is reflexive. I am aware of the opposite implication as presented by Hartshorne's "Stable reflexive sheaves and in the book of Okonek-Schneider-Spindler, but this implication is new to me.

(If it makes a difference, Stromme also knows $pd(F) \leq 1$ and $rk F=2$; furthermore, $F$ is a universal family of rank 2 bundles on the projective plane)

Does anyone know an explanation or reference for this fact?

I am reading "Ample divisors on fine moduli spaces on the Projective plane" by Stromme. In the proof of Proposition 2.4, he seems to claim that if $E$ is a torsion free sheaf of projective dimension at most 1 on a smooth variety and the singular locus of $E$ has codimension at least $3$, then $E$ is reflexive. Of course the converse is true without the projective dimension assumption.

Does anyone know an explanation or reference for this fact?