Arnold's chord conjecture for Legendrian submanifolds of the contact sphere

Let $\alpha$ be a contact form on the standard contact sphere $(S^{2n-1},\xi_{\mathrm{std}})$. A Reeb chord of a closed Legendrian submanifold $\Lambda$ is a Reeb trajectory of $\alpha$ whose endpoints lie on $\Lambda$. Arnold's chord conjecture. Every closed Legendrian submanifold $\Lambda$ of $S^{2n-1}$ has a Reeb chord. The conjecture asserts the existence of Reeb chords for arbitrary contact forms on the standard contact sphere; it was confirmed by Mohnke.