I hadn't realised is that neutrinos have never been observed going slower than light. If they had been observed going slower than light, then finding them also going faster would be absurd, since it would require infinite energy. But if they are always tachyons then them travelling faster than c is much less problematic.

However I don't see how this explains the neutrinos from the supernova. In the paper it says that higher energies correspond to lower speeds (due to imaginary mass). The ones at CERN are 17GeV, whereas the ones from the supernova were 6.7MeV. But the difference in time for the supernova was proportionately smaller than that for the CERN neutrinos.