The local Hamiltonian nondisplacement conjecture

Let $L$ be a displaceable Lagrangian in a symplectic manifold $M$, and let $d_{C^0}$ be the $C^0$ distance on Hamiltonian diffeomorphisms. Local Hamiltonian nondisplacement conjecture. There exists $\delta>0$ such that, whenever $\phi$ is a Hamiltonian diffeomorphism of $M$ satisfying

$$d_{C^0}(\mathds{1},\phi)<\delta,$$

then

$$L\cap\phi(L)\neq\emptyset.$$

This is a local $C^0$ rigidity statement for Hamiltonian images of a displaceable Lagrangian. The paper presents it as a conjectural variant of local nondisplacement; its general validity is not established in the supplied text.