Is there a mode stability paradox for neutrino perturbations of Kerr black holes?

Adopting the notation of Teukolsky and Press, we derive the connection relation for asymptotic solutions of the massless Dirac equation on a Kerr background. We show that, unlike bosonic fields, the connection relation for massless Dirac fields (neutrino) provides a rigorous proof of mode stability. The same relation also implies that every incoming mode can be absorbed by the black hole or there is no superradiance. Recent works on overspinning black holes have shown that this can lead to the formation of naked singularities. We argue that the fact that both the mode stability of the black hole under neutrino perturbations and the instability of the event horizon (therefore the instability of the black hole) can be derived from the same connection relation leads to a paradox. In other words mode, stability implies event horizon instability as far as neutrino perturbations are concerned.