Spectral bounded isometry conjecture

Let $(M,\omega)$ be a symplectic manifold, let $\operatorname{Symp}_0(M,\omega)$ be its identity component, and let $\operatorname{Ham}(M,\omega)$ be the Hamiltonian subgroup. Let $\operatorname{BI}_0(M,\omega,\gamma)$ denote the subgroup of symplectomorphisms bounded with respect to the spectral norm $\gamma$. Spectral bounded isometry conjecture.

$$\operatorname{BI}_0(M,\omega,\gamma)=\operatorname{Ham}(M,\omega)\qquad\text{for all }(M,\omega).$$

This is the spectral-norm analogue of the bounded isometry conjecture for the Hofer norm. The source formulates it as a conjecture after noting that the Hofer version is very hard and that partial results are known.