I recommend accepting this manuscript, provided the following points are clarified:
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.