'Tired light' appears to be an umbrella term for various disparate theories that are only related in their attempt to provide an alternative explanation for red shift. If this particular hypothesis, which is not explicitly cosmological, is validated in terrestrial experiments, I think its cosmological implications would be unavoidable.

> is validated in terrestrial experiments

They didn't run any experiments. They only have some a theoretical model and simulations in the computer.

The main part of the article, that says that the momentum of the light is split in the photon and in the density wave looks good. (It almost looks obvious, but it's not my expertise area https://xkcd.com/793/ .)

The relation with the Hubble constant is totally unexpected. It is not in the abstract of the peer review article. (I can't read the full text.) So it's probably only a declaration for the press release. Take it with a grain of salt.

The article like nearly every physical article has its preprint available on the arxiv.

https://arxiv.org/abs/1603.07224

note: "on the arxiv" and not "on arxiv". The name puns on x => greek letter chi.

If it's proven true, though, would that mean the "expanding universe" theory would be disproved as well?

Not necessarily. It might be that this effect is real (this was a computational simulation so an experimental verification is needed.) and needs to be accounted when doing cosmology. But it might very well account just a part of the redshift, and the remainder then needs to be accounted by something else, the default explanation being the expanding universe.

Edit: A clarification: I'm not an expert.

Without an expanding universe, one problem is "How does everything form, given that gravity exists?"

No expansion = everything falls together.

not if the universe is flat and simply connected (i.e, infinite): https://physics.stackexchange.com/questions/109063/how-come-...