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.
The treatment involves quantum corrections which are treated by calculating the Wigner representation of the density matrix and I am not aware of the use of such an extensive use of Wigner representation to describe atom interferometry experiments.