Let $f:X \to C$ be a flat, projective morphism of noetherian schemes with $C$, an irreducible curve. Suppose that there exists a point $o \in C$ such that the fiber $f^{-1}(o)$ is Fano. Is it true that in this case, every fiber of $f$, $f^{-1}(t)$ is Fano for all $t$?

N.B. If necessary, one can assume that $f$ is smooth, but if such a statement holds without this assumption, it will be more interesting.