  Let $\Lambda, \Lambda'$ be a pair of closed Legendrian submanifolds in a
closed contact manifold $(Y, \xi = Ker(\alpha))$ related by a Legendrian
cobordism $W\subset (\mathbb{C}\times Y, \tilde{\xi}=Ker(-y dx +\alpha))$. In
this note, we show that in the hypertight setting, if $\Lambda$ intersects a
closed, weakly exact or monotone pre-Lagrangian $P\subset Y$ for reasons of
Floer homology, then so does $\Lambda'$.
