The infinite spectral-diameter conjecture for symplectic manifolds

Let $(M,\omega)$ be a compact symplectic manifold, and let $\gamma$ be the Floer-theoretic pseudometric on the universal cover of $\mathrm{Ham}(M)$. Assume that $\omega$ vanishes on $\pi_{2}(M)$. Infinite spectral-diameter conjecture. The diameter of $\gamma$ is infinite. This folklore conjecture concerns the large-scale geometry of Hamiltonian diffeomorphism groups and is motivated by the spectral pseudometric from Hamiltonian Floer theory. Its resolution is not specified in the source.