Spectrally-large scale geometry in cotangent bundles

In this paper, we prove that the ${\rm Ham}$-orbit space from a fiber of a large family of cotangent bundles, as a metric space with respect to the Floer-theoretic spectral metric, contains a quasi-isometric embedding of an infinite-dimensional normed vector space. The same conclusion holds for the group of compactly supported Hamiltonian diffeomorphisms of some cotangent bundles. To prove this, we generalize a result, relating boundary depth and spectral norm for closed symplectic manifolds in Kislev-Shelukhin's recent work, to Liouville domains. Then we modify Usher's constructions (which were used to obtain Hofer-large scale geometric properties) to achieve our desired conclusions.