Closed-disk conjecture for the q-deformed Farey tessellation

Let $L_q$ be the proper subset of $\mathbb{H}$ covered by the $q$-deformed Farey tessellation. Its boundary satisfies $\partial\overline{L}_q=\mathbb{Q}_q\sqcup\mathbb{I}_q$, where $\mathbb{Q}_q$ is the union of the $q$-deformed rational curves and $\mathbb{I}_q$ is the set of $q$-deformed irrational numbers; moreover, this boundary is homeomorphic to $\mathbb{R}\cup\{\infty\}$. Closed-disk conjecture. The closure $\overline{L}_q$ of the $q$-deformed Farey tessellation is homeomorphic to a closed disk. The proposition preceding the conjecture establishes the open-disk structure and identifies the boundary; the conjecture asserts the corresponding global compactification.