Transcendence of simple geodesics in finite modular covers

Let $\Gamma=\operatorname{PSL}_2(\mathbb{Z})$, let $\Gamma^\prime\subset\Gamma$ be a finite-index subgroup corresponding to a finite cover $\mathbf{M}^\prime\to\mathbf{M}$, and let

$$\mathcal{G}(\mathbf{HP})=\mathbb{R}\operatorname{\mathbf{P}}^1\times\mathbb{R}\operatorname{\mathbf{P}}^1\setminus\mathrm{diagonal}$$

be the space of geodesics of the hyperbolic plane. For a geodesic $\xi=(\xi^-,\xi^+)\in\mathcal{G}(\mathbf{HP})$, let $\xi^\prime$ be its projection modulo $\Gamma^\prime$ to $\mathcal{G}(\mathbf{M}^\prime)$. A geodesic is simple when it has no transverse self-intersection in $\mathbf{M}^\prime$. Transcendence of simple geodesics in finite modular covers. If $\xi^\prime$ is simple, then $\xi^+\in\mathbb{R}\operatorname{\mathbf{P}}^1$ is either rational, quadratic, or transcendental. The bounded-cusp-excursion lemma shows that simple geodesics have bounded continued-fraction digits, so this conjecture is a finite-cover analogue of Khintchine's conjecture; it remains open.