Conjecture on geodesic-covering numbers of geometrically finite Fuchsian groups

Let $\Gamma$ be a discrete subgroup of $\mathrm{PSL}_2(\mathbb{R})$ whose fundamental domain has finitely many sides; such a group is geometrically finite. Geometrically finite geodesic-covering conjecture. The geodesic-covering number of $\Gamma$ is finite.

This generalizes the preceding finite-index modular-subgroup conjecture and is supported in the source by examples with finite geodesic-covering number, including a translation group and finite subgroups. The general claim remains unresolved there.