Just for the record, this does not need CM. The following is true:

>Let $Z$ be an excellent scheme that admits a dualizing complex. Then $ω_Z$ is torsion-free and $S_2$ on $Z$. If in addition Z is irreducible, then $ω_Z$ is a reflexive $\mathscr O_Z$ -module.

If there is interest in this, I will add a proof.