Let $X$ be a Fano manifold of Picard number one. It is known that $X$ is rationally connected. A component $K$ of rational curves in $X$ is called a minimal rational component if the evaluation map $\mathbb{P}^1 \times K \to X$ is dominant and the degree of the curves in $K$ is minimal with this property. A curve in such a component is called a minimal rational curve.

Question: Is it true that any two points $X$ can be connected by a chain of minimal rational curves?