Pair correlation of hyperbolic lattice angles

Let $\omega$ be a point in the upper half plane, and let $\mathrm{\Gamma }$ be a discrete, finite covolume subgroup of ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{R})$. We conjecture an explicit formula for the pair correlation of the angles between geodesic rays of the lattice $\mathrm{\Gamma }\omega$, intersected with increasingly large balls centered at $\omega$. We prove this conjecture for $\mathrm{\Gamma }={\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$ and $\omega$ an elliptic point.