Let $X$ be a projective, irreducible variety over an algebraically closed field (of characteristic zero) which is rationally connected. Is it true that any open dense subvariety of $X$ is rationally connected? If so, can we say that same for rationally chain connectedness?

By rationally (chain) connected we do not assume properness (similar to the definition in Kollar's "Rational curves on algebraic varieties"). We simply ask that two general points in the variety are connected by a (chain of) rational curve.

Any idea/reference is most welcome.