The referee implies that the leading term is not a quantum correction owing to the cancellation of a numerator proportional to hbar and a denominator in the spatial phase factor that is also proportional to hbar. This is, is some respect, a question of semantics, and has a long history in discussions of quantum effects in atom interferometry. Basically this term is a Doppler shift, which is clearly of classical origin, with no need to impose arguments related to cancellations of hbars in numerators and denominators. In the approach we follow using the Wigner representation, there are no numerators and denominators; the leading term has no hbar.
Moreover, the authors call this effect a quantum correction probably because it is proportional to hbar but it would be more clear to explain that, the leading term of the gravitation phase being independent of hbar because of a cancellation (hbar appears in the numerator and denominator of the quantity giving the phase), the term linear in gamma is also proportional to the recoil velocity which itself is proportional to hbar.