  In many of the approximate functionals in one-body reduced density matrix
(1RDM) functional theory, the approximate two-body reduced density matrix
(2RDM) in the natural orbital representation only depends on the natural
occupation numbers. In Phys. Rev. A 92, 012520 (2015) Wang and Knowles
initialised a discussion of to what extent this simplification is valid, by
introducing two different H$_4$ geometries with identical natural occupation
numbers, but different 2RDMs. Gritsenko has argued that this feature is due
symmetry [Phys. Rev. A 97, 026501 (2018)]. This work aims to contribute to the
discussion on the following points: 1) one should rather speak of
symmetry-restricted variants of the universal functional, than saying that the
universal functional is symmetry dependent; 2) the unitary invariance of
degenerate NOs can lead to large deviations in the 2RDM elements, especially
the phase of the NOs; 3) symmetry-restricted functionals are constructed for
the H$_4$ geometries considered by Wang and Knowles, whose structure could
serve as guide in the construction of approximate 1RDM functionals.
