  Measuring core-collapse supernova neutrinos, both from individual supernovae
within the Milky Way and from past core collapses throughout the Universe (the
diffuse supernova neutrino background, or DSNB), is one of the main goals of
current and next generation neutrino experiments. Detecting the heavy-lepton
flavor (muon and tau types, collectively $\nu_x$) component of the flux is
particularly challenging due to small statistics and large backgrounds. While
the next galactic neutrino burst will be observed in a plethora of neutrino
channels, allowing to measure a small number of $\nu_x$ events, only upper
limits are anticipated for the diffuse $\nu_x$ flux even after decades of data
taking with conventional detectors. However, paleo-detectors could measure the
time-integrated flux of neutrinos from galactic core-collapse supernovae via
flavor-blind neutral current interactions. In this work, we show how combining
a measurement of the average galactic core-collapse supernova flux with paleo
detectors and measurements of the DSNB electron-type neutrino fluxes with the
next-generation water Cherenkov detector Hyper-Kamiokande and the liquid noble
gas detector DUNE will allow to determine the mean supernova $\nu_x$ flux
parameters with precision of order ten percent.
