# Restriction of reflexive sheaf to fibers of a family

Let $X$ be a smooth projective variety over complex numbers. Let $f:Y\rightarrow Z$ be a flat family of divisors of $X$ parameterized by a smooth variety $Z$. Suppose $E$ is a reflexive sheaf on $Y$ flat over $Z$. Consider the restriction to the fibers $E|_{f^{-1}(z)}$. Do they continue to be reflexive, or at least torsion-free? Is there a non-empty open subset of $Z$ where this happens?

• There was some question here recently on this subject. – Sasha Aug 17 '16 at 22:07