Pick $n\ge 3$ non-vertical lines $\mathscr{L}:=\{\ell_1,\ldots,\ell_n\}$ in the plane which are pairwise non-parallel, and they are not all concurrent in a single point.

Question. Does there exist a vertical line $L$ which has $n-1$ or $n-2$ intersections with $\mathscr{L}$?

Partial note: By using a dual result due to Ungar here one can see that there are at least $n-1$ vertical lines $L$ which have at most $n-1$ intersections with $\mathscr{L}$