Let $X$ be a normal projective variety which has only terminal singularities. Let $\Omega^1_X$ be the Kahler differential sheaf on $X$ and $\omega_X$ be the dualizing sheaf on $X$. For a coherent sheaf $F$ on $X$, let $F^{* *}$ be its double dual.

**Question** Are ${\rm Ext}^2 ((\Omega^1_X)^{* *}, \mathcal{O}_X)$
and
${\rm Ext}^2 ((\Omega^1_X)^{* *} \otimes \omega_X, \omega_X) $ isomorphic?

If $X$ is Gorenstein, they are isomorphic. I want to know the situation when $X$ is non-Gorenstein.