Arnold–Givental conjecture for real Lagrangian fixed point sets

A real structure on a symplectic manifold $(M,\omega)$ is an antisymplectic involution $\rho\colon M\to M$ satisfying

$$\rho^2=\operatorname{id}|_M,\qquad \rho^*\omega=-\omega.$$

Let

$$L:=\operatorname{Fix}(\rho)\subset M$$

be its fixed point set, which is a (possibly empty) Lagrangian submanifold. The triple $(M,\omega,\rho)$ is a real symplectic manifold.

Arnold–Givental conjecture. Assume that $(M,\omega,\rho)$ is a closed real symplectic manifold and $\phi_H$ is a Hamiltonian diffeomorphism such that $L$ and $\phi_H(L)$ intersect transversally. Then

$$\#(L\cap\phi_H(L))\geq\sum_{i=0}^{\operatorname{dim}(L)} b_i(L;\mathbb{Z}_2).$$

The conjecture is a Hamiltonian nondisplaceability statement: it predicts that every transverse Hamiltonian image of the real Lagrangian fixed point set has at least the total number of $\mathbb{Z}_2$-Betti numbers of $L$ intersection points. Its resolution status is not specified in the supplied text.