Arithmetic holonomy conjecture for maximal volume ideal polyhedra

Let $S$ be the surface associated with a maximal volume ideal polyhedron, and let

$$\rho:\pi_1(S)\to\mathrm{PSL}(2,\mathbb{R})$$

be its holonomy representation. An arithmetic Fuchsian group is a subgroup of $\mathrm{PSL}(2,\mathbb{R})$ commensurable with the group of units in a quaternion algebra over a totally real number field. Arithmetic Holonomy Conjecture. The image of $\rho$ is commensurable with an arithmetic Fuchsian group. Rational dihedral angles make the shape parameters roots of unity and place the holonomy traces in cyclotomic fields, which is suggestive but does not by itself prove arithmeticity; the conjecture is open.