The unbounded autonomous Hofer distance conjecture

Let $X$ be a closed symplectic manifold. Define

$$\operatorname{aut}(X)=\sup_{\phi\in \operatorname{Ham}(X,\omega)}d_H(\phi,\operatorname{Aut}(X)),$$

where $d_H$ is Hofer's metric and $\operatorname{Aut}(X)$ denotes the autonomous Hamiltonian diffeomorphisms. Unbounded autonomous Hofer distance conjecture. For every closed symplectic manifold $X$,

$$\operatorname{aut}(X)=+\infty.$$

The conjecture predicts that autonomous Hamiltonian diffeomorphisms are unboundedly far from some Hamiltonian diffeomorphisms in Hofer's metric. The paper notes that this is known for products $\Sigma_g\times M$ under the hypotheses of the cited theorem, where $\Sigma_g$ is a closed oriented surface of genus $g\geq 4$ and $M$ is a closed symplectically aspherical manifold; the general case remains open.