We reexamine the possibility of reconstructing the initial fluxes of supernova neutrinos emitted in a future core-collapse galactic supernova explosion and detected in a Megaton-sized water Cherenkov detector. A novel key element in our method is the
inclusion, in addition to the total and the average energies of each neutrino species, of a pinching parameter characterizing the width of the distribution as a fit parameter. We uncover in this case a continuous degeneracy in the reconstructed parameters of supernova neutrino fluxes at the neutrinosphere. We analyze in detail the features of this degeneracy and show how it occurs irrespective of the parametrization used for the distribution function. Given that this degeneracy is real we briefly comment on possible steps towards resolving it, which necessarily requires going beyond the setting presented here.
