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Do Neutrino Oscillations Conserve Energy

If neutrinos have mass, they can be at rest. If neutrinos can oscillate between types with different masses, then Einstein’s relation E = mc [1], where c is the speed of light in vacuum, implies that the (rest) energy of a neutrino changes during a mass oscillation. Does the phenomenon of neutrino-mass oscillation imply that energy is not conserved, or (if energy is always conserved) are claims of such oscillation a hoax?