Algebraic-cycle conjecture for simple geodesics on arithmetic curves

Let $F\subset \operatorname{PSL}_2(\mathbb{R})$ be a lattice such that the complex algebraic curve $S=F\backslash\mathbb{H}$, or equivalently its Jacobian, is defined over $\overline{\mathbb{Q}}$. Fix a simple geodesic $\alpha\subset S$ and let $\widetilde{\alpha}\subset\mathbb{H}$ be its lift, an $F$-invariant set of geodesics. Algebraic-cycle conjecture. The following three conditions are equivalent: $\alpha$ is closed; $\alpha$ contains an algebraic point of $S$; and the endpoints $\partial\widetilde{\alpha}\subset\mathbb{R}\mathbb{P}^1$ are algebraic. For a simple closed curve, the endpoints of its lifts are quadratic over the invariant trace field $\K=\mathbb{Q}\{\operatorname{disc}(A)\mid A\in F\}$. This is a proposed Schneider-type algebraic/transcendence criterion for geodesics on arithmetic curves; the source gives no resolution status.