The Flux Conjecture

Let $(M,\omega)$ be a compact symplectic manifold. Write $\operatorname{Symp}(M,\omega)_0$ for the connected component of the identity in the symplectomorphism group, and let $\operatorname{Ham}(M,\omega)$ be the Hamiltonian diffeomorphism group.

Flux Conjecture. The subgroup $\operatorname{Ham}(M,\omega)$ is $C^1$ close in $\operatorname{Symp}(M,\omega)_0$; equivalently, it is $C^1$-closed there.

This is a classical conjecture attributed in the source to McDuff. Its formulation concerns the topology of Hamiltonian diffeomorphisms inside the identity component of the symplectomorphism group; the supplied text does not state whether it has been resolved.