Completeness of the higher-level Bethe-equation parametrization at low temperature

Let the XXZ chain be in the antiferromagnetic massive regime at finite magnetic field and sufficiently low temperature. The quantum transfer matrix has eigenstates, and the higher-level Bethe equations provide solutions describing particle-hole excitations. Completeness conjecture. For low temperatures at finite magnetic field every eigenstate of the quantum transfer matrix is parameterized by one of the solutions of the higher-level Bethe equations. This implies that all eigenstates can be interpreted in terms of particle-hole excitations. The authors support this belief with finite-Trotter-number numerical calculations and the free-fermion picture, but completeness is not established analytically.