No. From the paper linked above [0]:
reply> The phase of free fall is predicted in a purely quantum manner to have a dependence m/6 g^2T^3/ℏ + gmzT on the free-fall time T, where m is the mass of the object, g is the gravitational acceleration relative to the surface of Earth, and z in the spatial coordinate in the direction of gravity. This prediction follows the calculated phase accumulated by an object accelerating in a linear potential, and has been made starting from almost one hundred years ago by Darwin, Kennard and others.
Apparently the phase shift is derivable from just adding a linear potential term mgz to the Hamiltonian.