Why can't Wetterich's “Universe without expansion” be tested compared to standard cosmological redshift?

Question

I have read on several sites that this is a theory that cannot be tested. Why is this a theory that can not be tested? It seems to me that if the mass of particles is continually increasing, then the wavelength of a spectral line emitted by an atom would also be continually changing. This means that over a length of time (possibly not human time scale), if we could detect a shift in the emitted wavelength from an atom (stationary compared the observer), then we could determine if this interpretation is correct. We can obviously tell the difference in wavelength between the light emitted 10 billion years ago and that emitted today, so it should just be a matter of how fine our resolution is. The more narrowly we can resolve the wavelength, the shorter time span needed to make the measurement. If red shift is caused by movement relative to the observer, no red shift should be found. If red shift is caused by the mass change over time of the particles, red shift should be observable.