Weak Arnold–Givental conjecture on Hamiltonian nondisplaceability

Let $(M,\omega,\rho)$ be a closed real symplectic manifold, where $\rho$ is an antisymplectic involution, and let

$$L:=\operatorname{Fix}(\rho)\neq\varnothing$$

be its nonempty Lagrangian fixed point set.

Weak Arnold–Givental conjecture. The Lagrangian submanifold $L$ is Hamiltonianly nondisplaceable.

This is stated in the source as an immediate consequence of the Arnold–Givental conjecture, since the conjectural intersection lower bound is positive when $L$ is nonempty. The supplied text does not establish either conjecture, so the status is recorded as open.