Let $X$ be a smooth Fano variety of Picard number one. Let $Y \subset X$ be a closed subvariety. Let $d$ be an integer such that there is a dominating component $K_d$ of rational curves of degree $d$ in $X$. In other words, the family $K_d$ of curves covers $X$.

Question: For all such positive integer $d$, does there exist a rational curve $l \in K_d$ passing through a general point $p \in Y$ which lie in $Y$?

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