Such a theory would help to explain the apparent shortfall of neutrinos coming from the Sun.
Some tests find tentative but ambiguous evidence for oscillation.
They study neutrinos made when cosmic rays from outer space strike the upper atmosphere. Some neutrinos, those made overhead above Japan, travel about 20 km or so before entering the underground detector. Other neutrinos, those made in the atmosphere on the far side of the globe, have a travel path of 12,700 km into the detector. A 1-GeV muon neutrino seems to oscillate every few hundred miles.