  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.
