Neutrino beam experiments, like MiniBoone and T2K, are setup to look for evidence of neutrino oscillations that occur between neutrino creation and the detector. Are the neutrino beams well defined enough to measure the change in momentum as Earth's gravitational field interacts with the beam (i.e. can we measure the bend in the neutrino beam)? If that has been done, how far are we from using the bend in the beam to detect deviations caused by neutrino mass (which are, admittedly, of order $m_\nu^2 / E_\nu^2$ in natural units, and therefore tiny)? Do the prospects get significantly better in space where multiple beams can be fired past a much larger body (e.g. firing a laser and neutrino beam past the sun, though fantastic collimation of both would have to be achieved)?
Sorry for letting the question get a little hair brained at the end there. It's just fun to imagine the possibilities. :)
Preemptively - not a duplicate of "Neutrino path bent by gravity? [duplicate]", or the questions it is a duplicate of, because I'm asking about (the prospect of) experimental evidence, not the theory.