[photon]

Even 1m/s off would have huge implications, it would mean space-time and gravity behave fundamentally different than what we understood, invalidating our understanding of tons of things, like the equivalence principle. At a simpler level different colours of light would be effected differently (since they have different energies), so if nothing else this effect would scatter light, which we would probably have noticed by now.

I've been reading around a bit, I haven't seen anything yet from my usual sources, but I still don't understand what he's proposing is actually going on.

If a photon is going along, spontaneously becomes an electron positron pair, that pair would have the same energy and momentum as the photon, so would be affected by gravity identically (since gravity acts on what's there including momentum, not on the rest mass of things). So the paper must be proposing something else.

I looked at the paper, but when it starts involving math with Hamiltonians I'm in way over my head, but the stuff I've read so far seems to suggest he's not using the equations properly.

As always best to take new stuff glommed onto by media with a grain of salt.