Reduction of G-SPT classification to a finite subgroup

Determine whether for every compact Lie group G there exists a finite subgroup H ≤ G such that the set of one-dimensional G-symmetry protected topological phases coincides with the set of H-symmetry protected topological phases.

Background

For many continuous groups, H^2(G,U(1)) is finite. Physically, certain continuous symmetries can sometimes be reduced to discrete subgroups without losing SPT distinctions (e.g., SO(3) versus its Z2×Z2 dihedral subgroup).

This conjecture asks whether such a reduction always exists in one dimension, which would simplify classification and computation of SPT invariants for compact Lie group symmetries.