Rational Angle Conjecture for maximal volume ideal polyhedra

Let $\mathcal{T}$ be a realizable triangulation of the sphere, and consider the volume-maximizing ideal polyhedron with combinatorial type $\mathcal{T}$. Rational Angle Conjecture. All of its dihedral angles have the form

$$\frac{p\pi}{q}$$

for integers $p$ and $q$. The optimization problem has no built-in preference for rational angles, and the phenomenon has been verified for the globally maximal configurations with $n=4,\ldots,12$, but remains unproved for every realizable triangulation.