  In this article, we study the on-shell production of low-mass vector
mediators from neutrino-antineutrino coalescence in the core of proto-neutron
stars. Taking into account the radial dependence of the density, energy, and
temperature inside the proto-neutron star, we compute the neutrino-antineutrino
interaction rate in the star interior in the well-motivated
$U(1)_{L_{\mu}-L_{\tau}}$ model. First, we determine the values of the coupling
above which neutrino-antineutrino interactions dominate over the Standard Model
neutrino-nucleon scattering. We argue that, although in this regime a
redistribution of the neutrino energies might take place, making low-energy
neutrinos more trapped, this only affects a small part of the neutrino
population and it cannot be constrained with the SN 1987A data. Thus, contrary
to previous claims, the region of the parameter space where the
$U(1)_{L_{\mu}-L_{\tau}}$ model explains the discrepancy in the muon anomalous
magnetic moment is not ruled out. We then focus on small gauge couplings, where
the decay length of the new gauge boson is larger than the neutrino-nucleon
mean free path, but still smaller than the size of proto-neutron star. We show
that in this regime, the on-shell production of a long-lived $Z'$ and its
subsequent decay into neutrinos can significantly reduce the duration of the
neutrino burst, probing values of the coupling below ${\cal O}(10^{-7})$ for
mediator masses between 10 and 100 MeV. This disfavours new areas of the
parameter space of the $U(1)_{L_{\mu}-L_{\tau}}$ model.
