Prove the Galois-orbit relation for higher-power monodromy TQFTs

Prove that the modular data of the three-dimensional TQFTs obtained from higher-power monodromy traces $\operatorname{Tr}\Phi^n$ form a Galois orbit for every $(A_1,G)$ Argyres–Douglas theory, rather than only for the examples whose BPS quivers have at most six nodes.

Background

The monodromy operator has finite periodicity, so the TQFTs associated with different powers of the monodromy form a finite family. The paper computes modular data for a range of theories and finds that the data form Galois-related families in all cases examined.

The authors present this relation as a belief rather than a theorem and restrict their explicit verification to theories with BPS quivers having at most six nodes. A general proof, or an extension to all relevant $(A_1,G)$ theories and monodromy powers, remains unresolved.