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?**