I am currently working on some problems related to Grassmann manifolds and eventually come to the following question.

Let $S$ be a subset of $G_1(\mathbb{R}^n)$ such that any element of $G_n:=G_{n-1}(\mathbb{R}^n)$ contains a line in $S.$ Is it true that there exists a hyperplane in $G_{n-1}$ that is completely covered by $S$ (that is, every line contained in that hyperplane belongs to $S$)?

Has this problem ever been discussed before? So far I could not come up with any good idea for it, at least not by "geometric arguments".