I think that the uncertainty about the model and assumptions is already larger than 1/1000, especially given the recent failed "faster-than-light neutrinos" LHC experiment. So while 5 sigmas would be nice, I would assume that the physicists over there are busy double-checking everything, and trying to reproduce the result, rather than improve their "sigma-score".
Can any physicist confirm? I'm not all that familiar with how these things work. Furthermore, if they perform once more the experiment and get the same 3-sigma result, wouldn't that compound into a >3-sigma total?