  We study the following rigidity problem in symplectic geometry:can one
displace a Lagrangian submanifold from a hypersurface? We relate this to the
Arnold Chord Conjecture, and introduce a refined question about the existence
of relative leaf-wise intersection points, which are the Lagrangian-theoretic
analogue of the notion of leaf-wise intersection points defined by Moser. Our
tool is Lagrangian Rabinowitz Floer homology, which we define first for
Liouville domains and exact Lagrangian submanifolds with Legendrian boundary.
We then extend this to the `virtually contact' setting. By means of an
Abbondandolo-Schwarz short exact sequence we compute the Lagrangian Rabinowitz
Floer homology of certain regular level sets of Tonelli Hamiltonians of
sufficiently high energy in twisted cotangent bundles, where the Lagrangians
are conormal bundles. We deduce that in this situation a generic Hamiltonian
diffeomorphism has infinitely many relative leaf-wise intersection points.
