The extension conjecture for the Hofer metric on symplectomorphism groups

Let $(M,\omega)$ be a symplectic manifold, and let $\operatorname{Ham}(M,\omega)$ and $\operatorname{Symp}_0(M,\omega)$ denote its Hamiltonian group and the identity component of its symplectomorphism group, respectively. The Hofer metric is the bi-invariant metric on $\operatorname{Ham}(M,\omega)$.

Extension conjecture. For all symplectic manifolds $(M,\omega)$ such that

$$\operatorname{Symp}_0(M,\omega)\ne\operatorname{Ham}(M,\omega),$$

the Hofer metric on $\operatorname{Ham}(M,\omega)$ does not extend to a bi-invariant metric on $\operatorname{Symp}_0(M,\omega)$.

This is presented as a more accessible conjecture than the bounded isometry conjecture. The preceding theorem shows that an unbounded symplectomorphism obstructs such an extension, while the conjecture claims that no extension exists whenever the identity component is strictly larger than the Hamiltonian group.