I was thinking of how Lorentz invariance was historically arrived at: From Maxwell's equations. If the photon has mass, then presumably Maxwell does not exactly describe its behaviour (although with the current upper bound it will be a very good approximation); but the underlying massless gauge field may still follow Maxwell.
First we may clarify what is exactly meant by "following Maxwell". For example in electrodynamics (weak interaction switched off) there is interaction between electron field and photons. Is this Maxwell? Classical Maxwell equations include the interaction of electromagnetic field and current and charge densities, but they don't include equation of motion for the charges. Nevertheless, we can say that in quantum electrodynamics
http://www.nature.com/news/2011/110922/full/news.2011.554.html
http://arxiv.org/abs/1109.4897v1
http://usersguidetotheuniverse.com/?p=2169
http://news.ycombinator.com/item?id=3027056
Perhaps the end of the era of the light cone and beginning of the era of the neutrino cone? I'd be curious to see your probability estimates for whether this theory pans out. Or other crackpot hypotheses to explain the results.