The bounded isometry conjecture for symplectomorphism groups

Let $(M,\omega)$ be a closed symplectic manifold with symplectic form $\omega$. Let $\operatorname{Ham}(M,\omega)$ be its Hamiltonian group and let $\operatorname{Symp}_0(M,\omega)$ be the identity component of its symplectomorphism group. For $\phi\in\operatorname{Symp}(M,\omega)$, define

$$r(\phi):=\sup\{\rho([\phi,f])\mid f\in\operatorname{Ham}(M,\omega)\},$$

where $\rho$ is the Hofer norm on $\operatorname{Ham}(M,\omega)$ and $[\phi,f]:=\phi f\phi^{-1}f^{-1}$. A symplectomorphism is bounded if $r(\phi)<\infty$, and let $\operatorname{BI}_0(M,\omega)$ denote the set of bounded elements of $\operatorname{Symp}_0(M,\omega)$.

Bounded isometry conjecture. For all $(M,\omega)$,

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

The conjecture asserts that every bounded symplectomorphism in the identity component is Hamiltonian. It would imply that every non-Hamiltonian element is unbounded, providing an obstruction to extending the Hofer metric to a bi-invariant metric on the full identity component.