The clique number of unit quasi-disk graphs

For $\epsilon \in [0,1]$, a unit $\epsilon$-quasi-disk is a connected compact set $Q$ of the plane such that there exists a point $P$ such that $D(P,1-\epsilon) \subseteq Q \subseteq D(P,1)$, where $D(C,r)$ denotes the disk of centre $C$ and radius $r$. We prove that for any fixed $\epsilon>0$, the clique number problem on the class of intersection graphs of unit $\epsilon$-quasi-disks is NP-complete.