Integral but non-superintegral polyhedral circle packings

Classify the integral polyhedral circle packings that are not superintegral.

Background

Polyhedral circle packings arise from the Koebe–Andreev–Thurston construction. A packing is integral when all circles have integer curvatures, while it is superintegral when the larger superpacking also has integer curvatures.

The paper’s arithmeticity classification gives a complete classification of superintegral polyhedra, because superintegrality implies arithmeticity. It does not resolve the broader class of integral packings that fail to be superintegral; the authors note that quasi-arithmeticity, rather than arithmeticity, is relevant in this setting.