  In this paper we investigate a time-varying neutrino mass model, motivated by
the mild tension between cosmic microwave background (CMB) measurements of the
matter fluctuations and those obtained from low-redshift data. We modify the
minimal case of the model proposed by Dvali and Funcke (2016) that predicts
late neutrino mass generation in a post-recombination cosmic phase transition,
by assuming that neutrino asymmetries allow for the presence of relic neutrinos
in the late-time Universe. We show that, if the transition is supercooled,
current cosmological data (including CMB temperature, polarization and lensing,
baryon acoustic oscillations, and Type Ia supernovae) prefer the scale factor
$a_s$ of the phase transition to be very large, peaking at $a_s\sim 1$, and
therefore supporting a cosmological scenario in which neutrinos are almost
massless until very recent times. We find that in this scenario the
cosmological bound on the total sum of the neutrino masses today is
significantly weakened compared to the standard case of constant-mass
neutrinos, with $\sum m_\nu<4.8$~eV at 95\% confidence, and in agreement with
the model predictions. The main reason for this weaker bound is a large
correlation arising between the dark energy and neutrino components in the
presence of false vacuum energy that converts into the non-zero neutrino masses
after the transition. This result provides new targets for the coming KATRIN
and PTOLEMY experiments. We also show that the time-varying neutrino mass model
considered here does not provide a clear explanation to the existing
cosmological $\Omega_m$-$\sigma_8$ discrepancies.
