Unboundedness conjecture for the contactomorphism-group metric $d_{cc}$

Let $W$ be the completion of a Liouville domain, and let $\overline{\operatorname{Cont}}_{0,c}(W\times S^1)$ be the $C^0$-completion of the identity component of the group of compactly supported contactomorphisms of $W\times S^1$, equipped with the topology $\tau_{C^0}$. Let $d_{cc}$ denote the associated conjugation-invariant metric. Unboundedness conjecture. The metric $d_{cc}$ is unbounded on $\bigl(\overline{\operatorname{Cont}}_{0,c}(W\times S^1),\tau_{C^0}\bigr)$. This conjecture would establish that the candidate spectral norm $\gamma_{AM}$ is genuinely unbounded and would extend the paper's results concerning conjugation-invariant norms on contactomorphism groups; its resolution is not supplied here.