The infinite quasi-flat conjecture for Hamiltonian groups of Liouville domains

Let $(W,\omega)$ be a Liouville domain, and let ${\rm SH}^*(W,\omega)$ denote its symplectic homology. Equip $\operatorname{Ham}(W,\omega)$ with the metric $d_{\gamma}$. Infinite quasi-flat conjecture. If

$${\rm SH}^*(W,\omega)\neq 0,$$

then the metric space $(\operatorname{Ham}(W,\omega),d_{\gamma})$ contains a rank-$\infty$ quasi-flat. The conjecture is motivated by the preceding theorem for cotangent bundles and suggests that the phenomenon is governed by the Morse homology of the core, or Lagrangian skeleton, of a Liouville domain.