I have seen theories like this, and it is worth considering. However, despite the assumption of the author, there are a few items that may limit the application to cosmic redshift.
First, these equations are applicable in air or “slightly compressible liquids”, but the ideal vacuum of space is neither. Although there is no true vacuum, the empty parts between galaxies are largely elemental hydrogen and weakly interacting particles which would have a very narrow band filtering effect per this theory if any at all.
Second, the author claims that spectral lines are packets as opposed to single wavelengths. However, the very definition of a quantum leap is that each leap produces a single precise wavelength. It is a very pure frequency reference, and if there is an unequal treatment of each frequency in the spectrum, then the element would not be identifiable. That is unless it is the case that the redshift of the spectral lines is not equally transposed, but I do not expect that is the case or it would have been apparent since Hubble’s time.

Additionally, the few multi-messenger observations we can reference, including LIGO GW170817, show that longer wavelengths seem to experience propagation delay that increases with wavelength. This implies that, over parsecs, that lower frequency wave packets may in fact interact with intergalactic particles more frequently than higher frequencies. Therefore, the inverse of the theory presented would be observed.

I do think that there are more plausible alternatives to metric expansion that have yet to be considered. This one in particular allows for spectral lines to remain coherent while a linear shift over distance is observed.

Thank you for posting the link to that paper. I personally appreciate any opportunity to consider other theories.