The importance of detecting neutrinos from a Milky Way core-collapse supernova is well known. An under-studied phase is proto-neutron star cooling. For SN 1987A, this seemingly began at about 2 s, and is thus probed by only 6 of the 19 events (and only the $bar{ u}_e$ flavor) in the Kamiokande-II and IMB detectors. With the higher statistics expected for present and near-future detectors, it should be possible to measure detailed neutrino signals out to very late times. We present the first comprehensive study of neutrino detection during the proto-neutron star cooling phase, considering a variety of outcomes, using all flavors, and employing detailed detector physics. For our nominal model, the event yields (at 10 kpc) after 10 s -- the approximate duration of the SN 1987A signal -- far exceed the entire SN 1987A yield, with $simeq$250 $bar{ u}_e$ events (to 50 s) in Super-Kamiokande, $simeq$110 $ u_e$ events (to 40 s) in DUNE, and $simeq$10 $ u_mu, u_tau, bar{ u}_mu, bar{ u}_tau$ events (to 20 s) in JUNO. These data would allow unprecedented probes of the proto-neutron star, including the onset of neutrino transparency and hence its transition to a neutron star. If a black hole forms, even at very late times, this can be clearly identified. But will the detectors fulfill their potential for this perhaps once-ever opportunity for an all-flavor, high-statistics detection of a core collapse? Maybe. Further work is urgently needed, especially for DUNE to thoroughly investigate and improve its MeV capabilities.
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