Touch of Neutrinos on the Vacuum Metamorphosis: is the $H_0$ Solution
  Back?
  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 explanation 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. To estimate the effect on
cosmological observables we implement various extensions of the VM model in the
standard \texttt{CosmoMC} pipeline and establish which regions of parameter
space are empirically viable to resolve the $H_0$ and $S_8$ tensions. We find
that the likelihood analyses of the physically motivated VM model, which has
the same number of free parameters as in the spatially-flat $\Lambda$CDM model,
always gives $H_0$ in agreement with the local measurements (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. [Abridged]
