Lalonde–Polterovich bounded isometry conjecture

Let $(M,\omega)$ be a symplectic manifold. Write $\operatorname{Symp}_0(M)$ for the identity component of the symplectomorphism group, and let ${\rm BI}(M)$ be the normal subgroup of bounded symplectomorphisms. Define ${\rm BI}_0(M)={\rm BI}(M)\cap\operatorname{Symp}_0(M)$. The Hamiltonian diffeomorphism group is a subgroup of ${\rm BI}_0(M)$. Lalonde–Polterovich bounded isometry conjecture.

$${\rm BI}_0(M)={\rm Ham}(M,\omega).$$

This conjecture asks whether every bounded symplectomorphism in the identity component is Hamiltonian; it is the proposed extension of the Hofer metric from Hamiltonian diffeomorphisms to symplectomorphisms.