My only worry about this: what if they have, in fact, only measured the Earth's movement through the universe?
I'm reminded of the astronomers who were at a conference and who were all ready to announce that they'd found an exoplanet with a 1-year orbital period...and then just before their talk one of them realized that their calculations hadn't canceled the Earth's motion around the sun, and that they had, in fact, ended up measuring Earth's orbit in a very left-handed fashion.
The basis for this theory regarding alpha comes from measuring spectra, which is the same source for all of our information about anything we can't actually go stand on and kick the ground of. Big Bang, expanding universe, dark matter, dark energy, etc--all from measuring the spectra of various distant objects and determining where their absorption lines were. We're pretty sure that we're doing things correctly and getting the right info from the data.
But if alpha is varying non-symmetrically along one axis, how can you be sure it's not just due to the Earth's motion? There is no way for us to compare the Earth's motion to anything else in the universe, not on this scale. Since there are no fixed reference points, all we can do is point to a cloud of galaxies and say, "We're generally moving that-a-way."
I'd be a lot less suspicious if the axis didn't pass right through the Earth's core.
(Via Michael Flynn.)