Unbounded Hofer distance from autonomous Hamiltonian diffeomorphisms

Let $(M,\omega)$ be a closed symplectic manifold. Denote by $\operatorname{Ham}(M,\omega)$ the group of $C^\infty$-smooth Hamiltonian diffeomorphisms, equipped with Hofer's metric $d$, and let $\operatorname{Aut}\subset\operatorname{Ham}(M,\omega)$ be the set of autonomous Hamiltonian diffeomorphisms. Define

$$\operatorname{aut}(M,\omega):=\sup_{\phi\in\operatorname{Ham}(M,\omega)}d(\phi,\operatorname{Aut}).$$

Unboundedness conjecture. For every closed symplectic manifold $(M,\omega)$,

$$\operatorname{aut}(M,\omega)=+\infty.$$

This asserts that autonomous Hamiltonian diffeomorphisms can lie arbitrarily far from arbitrary Hamiltonian diffeomorphisms in Hofer's geometry. The supplied text does not indicate whether the statement is proved or remains open.