Suppose in the last step of a MMP, we obtain a Mori fiber space $f: X \to Z$, and let $F$ be a general fiber of $f$, then is the Picard number $\rho(F)$ of $F$ equal to $1$? Notice that the relative Picard number $\rho(X/Z)=1$ because I only assume to contract an extremal ray.
My feeling is that $\rho(F)$ may not be one, but I don't have a good example. Notice that if $X$ is toric, the fiber always has Picard number $1$.
I would also appreciate examples (if any) of fibration (with connect fibers) $f: X \to Y$ such that $\rho(X/Z)=1$ but the general fiber does not have Picard number $1$.