Lagrangian Poincaré recurrence conjecture

Let $(X,\omega)$ be a symplectic manifold, let $\psi\in\operatorname{Ham}(X,\omega)$, and let $L\subset(X,\omega)$ be a closed Lagrangian submanifold. Lagrangian Poincaré recurrence conjecture. There exists a sequence $k_i\to\infty$ of natural numbers such that

$$\psi^{k_i}(L)\cap L\neq\varnothing.$$

Furthermore, the density of the sequence $k_i$ is related to a symplectic capacity of $L$. The conjecture is disproved: counterexamples are known in all symplectic manifolds of dimensions at least six and in some symplectic manifolds of dimension four.