Sagittarius A-Star said:
the paper must be wrong in the lab-frame
Yes, in the sense that it assumes that its interpretation of "energy" is the
only viable one, which is not the case.
The paper is using "energy" to denote energy at infinity, which is indeed a constant of geodesic motion in any stationary spacetime; so on this interpretation it is true that the photon does not and cannot change energy as it propagates.
However, you are describing a different use of the term "energy", which is perfectly valid. See further comments below.
Sagittarius A-Star said:
The energy of a photon is frame-dependent.
More precisely, it is dependent on the spacetime relationship between the emitter and the receiver. You can actually construct a scalar invariant that describes the energy of a photon as measured by a particular emitter or receiver: it's just the inner product of the photon's 4-momentum with the receiver's 4-velocity. And if we pick an emitter and receiver at different altitudes in the gravitational field, and a photon propagates between them, the photon's 4-momentum stays the same (for the same reason that energy at infinity is a constant of the motion) but the receiver's 4-velocity is different from the emitter's 4-velocity, so the energy measured by the receiver will be different than the energy measured by the emitter.
Your particular
description of this process is frame-dependent, yes; it requires you to choose a local inertial frame that contains the emitter, the receiver, and the photon. But the invariant I described above can be constructed for
any emitter and receiver, even if they are much too far apart to both fit within the same local inertial frame (in which case your description could not even be applied).