Let $f:X \to T$ be a flat, projective morphism of noetherian schemes with $T$ an irreducible curve. Suppose that there exists a point $0 \in T$ such that the fiber $f^{-1}(0)$ is Fano.

Q.Is it true that in this case, for all $t$ near $0$, the fiber $f^{-1}(t)$ is Fano?

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

all$t \in C$. Even if $f$ is smooth, this will typically only hold for $t$ near 0. $\endgroup$