Tachyonic neutrinos: What do the data show?

In 1962 George Sudarshan and colleagues Deshpande and Bilaniuk suggested the possible existence of tachyons, superluminal particles that had $m^2<0.$ Neutrinos, of course, are the only known particles that might be tachyons, and while the values of their three masses are as yet unknown apart from upper limits, an important clue was discovered by Ramanath Cowsik in 1988, who used the neutrino data from supernova 1987A to infer two of the three neutrino masses. However, that result depended on the neutrino emissions from SN 1987A being near-simultaneous, i.e., within a time interval $\mathrm{\Delta }t<1s,$ which is contradicted by virtually all supernova neutrino emission models. Here it is explained why those neutrino emission models are probably mistaken, as are all the other arguments against one of the three neutrinos having $m^2<0.$