Even if you don't have spectral features your continuum isn't even, then this will cause the shape of the absorption line to shift and this could cause changes in the location of the peaks. Also the continuum spectra could be highly polarized causing other peaks to move.I don't understand- they are measuring the location of absorption peaks due to 'nearby' galaxies; what is the role of quasar gas dynamics as a source of error? It seems that the source does not have spectral features- at least, not in the spectral region they are using.
The other thing is that the lines could come from different parts of the galaxy. You could have one set of lines come from the galactic core. And another line coming from out in the disk. If these two different gas clouds are moving with respect to each other, you are going to get spurtious doppler shifts.
If there is something about the clouds that cause all of the numbers to be shifted systemically the same amount, then I don't see how any of the tests that present would rule that out. Something that bothers me about their data is that if you just draw a straight line through it, it doesn't end up at z=0,alpha=0Such as...?
I'm also not seeing how their systematics rules out a local (i.e. solar system effect).
Also just because it is in the solar system doesn't mean that it isn't interesting. There are some models of alpha variation in which alpha will change based on the locations of the earth....
One thing that they've done a good job doing is to try to establish that the effect isn't in the telescope. As long as it is outside the telescope, it's likely to be something interesting.