The origin of neutrino masses remains unknown. Both the vacuum mass and the dark mass generated by the neutrino interaction with dark matter (DM) particles or fields can fit the current oscillation data. The dark mass squared is proportional to the DM number density and, therefore, varies on the galactic scale with much larger values around the Galactic Center. This affects the group velocity and the arrival time delay of core-collapse supernovae (SN) neutrinos. This time delay, especially …