  With the entrance of cosmology in its new era of high precision experiments,
low- and high-redshift observations set off tensions in the measurements of
both the present-day expansion rate ($H_0$) and the clustering of matter
($S_8$). We provide a simultaneous solution of these tensions using the
Parker-Raval Vacuum Metamorphosis (VM) model with the neutrino sector extended
beyond the three massless Standard Model flavours and the curvature of the
universe considered as a model parameter. We constrain the parameter space
employing the following data sets: {\it (i)}~the cosmic microwave temperature
and polarization data from the Planck mission, {\it (ii)}~Baryon Acoustic
Oscillations (BAO) measurements, and {\it (iii)}~the Pantheon sample of
Supernovae type Ia. We find that the likelihood analyses of the physically
motivated VM model, which has the same number of free parameters than the
spatially-flat $\Lambda$CDM model, always gives a solution to the $H_0$ tension
(even when BAO or Pantheon data are included) at the price of much larger
$\chi^2$ than $\Lambda$CDM. The inclusion of massive neutrinos and extra
relativistic species quantified through two well known parameters $\sum
m_{\nu}$ and $N_{\rm eff}$, does not modify this result, and in some cases
improves the goodness of the fit. In particular, for the original VM+$\sum
m_\nu$+$N_{\rm eff}$ and the Planck+BAO+Pantheon dataset combination, we find
evidence for $\sum m_{\nu}=0.80^{+0.18}_{-0.22}~{\rm eV}$ at more than
$3\sigma$, no indication for extra neutrino species, $H_0=71.0\pm1.2$~km/s/Mpc
in agreement with local measurements, and $S_8=0.755\pm0.032$ that solves the
tension with the weak lensing measurements.
