Let $Y$ be a rationally connected variety over an algebraically closed field, and let $$\phi:X\dashrightarrow Y$$ be a rational fibration such that the general fiber of $\phi$ is rationally chain connected. Is it true that $X$ is rationally chain connected?
If we assume that the general fiber of $\phi$ is smooth and rationally connected can we conclude that $X$ is rationally connected?