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The evidence I am aware of is the tritium beta-decay end-point
uncertainty, which overlaps the negative mass squated region. The
uncertainty is large; after all, the mass-squared differences are no
measured to be of
the order of 10^{-5} eV, so if 1 neutrino is massless, or nearly so, the
beta-deecay end point uncertainty is huge in comparison.
I have just checked the LSD web-site, and I don't see any claim
for the data you are discussing. Where are these data reported?
Early neutrinos from a supernova event are more likely to be just
that; neutrinos that got out during the first stages. The main burst of
neutrinos takes a very long time to plough through the dense material that
collapses onto the center of the star, and then there are bounces. The
burst cannot be detected optically for a while because the light takes
even longer to escape.
If someone is making a serious claim for evidence of superluminal
particles I would expect it to be submitted to the ArXiV. Do you have a
citation?