Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$. Assume that the characteristic of $K$ is $0$ and of $k$ is $p>0$. Let $\pi:X \to \mbox{Spec}(R)$ be a flat, projective morphism with $X$ a regular scheme. Is it true that a general hypersurface of $H_X$ is a regular scheme, flat over $\mbox{Spec}(R)$?

If I understand correctly, this is true "locally" by a result of Flenner proven in Die Sätze von Bertini für lokale Ringe.