  Gauge non-invariance due to material truncation has recently been explored in
a number of contexts in strong-coupling QED. We show that the approach proposed
recently in Nature Physics 15, 803 (2019) rests on an incorrect mathematical
assertion and so does not resolve gauge non-invariance. It produces new
two-level models that are not equivalent in different gauges. The new
Coulomb-gauge model is inaccurate for the regimes considered in Nature Physics
15, 803 (2019), for which the multipolar-gauge quantum Rabi model is accurate.
The models analysed in Nature Physics 15, 803 (2019) do not result from the
argument provided in the main text, but instead from truncation within the
multipolar-gauge followed by the application of a truncated phase-invariance
principle. More generally, this principle can be applied following truncation
in any gauge and it yields an equivalence class of truncated models within the
gauge chosen. Equivalence classes belonging to different gauges are not
equivalent and the truncated phase-invariance principle does not provide an
argument to prefer a particular class. In general, the optimal class depends on
the physical situation, including the observables, the parameter regime, and
the number of field modes being considered. We also emphasise that
gauge-ambiguities are not synonymous with gauge non-invariance due to
approximations. Independent of material truncation subsystem predictions can be
vastly different depending on the theoretical definitions of the subsystems,
which are controlled by the gauge choice. This constitutes an example of
vector-space relativity that in no way contradicts gauge-invariance. However,
within ultrastrong-coupling regimes this relativity can no longer be ignored.
