By "every particle" do you mean like, every type of particle?

If you mean individual particles, fundamental particles don't really have distinct individual identities (as shown by fermi and bose statistics).

As for types of particles: Well, photons surely move at the speed photons move at.

Special relativity is derived from the assumption/observation that light travels at the same speed in all inertial reference frames, and generally that the laws of physics work the same in any inertial reference frame.

What you are proposing sounds pretty vague and unclear to me, but, is what you are trying to say compatible with this?

Probably no, because according to quantum mechanics, photons are indistinguishable in a way that affects experiments.

Before you figure that out, you will be surprised to discover that it's impossible to tell if the speed of light is the same in both directions, with only the average of a round trip being the speed of light.

See: https://en.wikipedia.org/wiki/One-way_speed_of_light

Anyway, reading that should help you answer your own question, or at least give you a lot more questions to ask.

What measurable predictions does this hypothesis make that would differ from the existing models?

Seems like they just added a linear gravitational potential term mgz to their hamiltonian and computed the phase change it would induce. They claim they experimentally confirmed the phase change. I didn't think that worked! They also claim its consistent with some kind of GR derivation, but I haven't looked at that.