Completeness of unit shape for other Galois groups

Determine whether the unit shape uniquely determines number fields up to isomorphism in the other Galois-group families listed for unit-rank-2 number fields.

Background

The paper proves conditionally that the unit shape is a complete invariant for totally imaginary non-CM $D_6$-sextic fields, while showing that it is generally too weak an invariant for CM fields. The remaining Galois-group families may exhibit different behavior.

The authors leave unresolved whether the completeness result extends to other groups in the rank-2 classification. Resolving this would characterize the distinguishing power of unit shapes across the broader range of admissible families.