The monotone Lagrangian cuplength conjecture

Let $L$ be a compact nondisplaceable monotone Lagrangian submanifold of a tame symplectic manifold $(M,\omega)$, let $\varphi\in\mathcal{H}am_c(M,\omega)$, and let $\mathbb{F}$ be the coefficient field used to define cuplength. The monotone Lagrangian cuplength conjecture. The number of intersection points of $\varphi(L)$ with $L$ is greater than or equal to the $\mathbb{F}$-cuplength of $L$. An affirmative answer would imply the corresponding fixed-point lower bound for Hamiltonian diffeomorphisms of closed monotone symplectic manifolds; the source presents the conjecture as open in general.