Conjecture on geodesic-covering numbers of finite-index modular subgroups

Let $\Gamma$ be a subgroup of $\mathrm{PSL}_2(\mathbb{Z})$ of finite index, and let its geodesic-covering number be defined using a fundamental domain. Finite-index modular subgroup conjecture. The geodesic-covering number of $\Gamma$ is finite.

The claim is motivated by the finite-union relationship between fundamental domains of a Fuchsian group and those of its finite-index subgroups. It is presented as an expected general finiteness property, with no resolution supplied in the source.