  We investigate a two-parameter extension of the $\Lambda_{\rm s}$CDM model by
allowing variations in the effective number of neutrino species $N_{\rm eff}$
and their total mass $\sum m_\nu$. Our motivation is twofold: (i) to examine
whether $\Lambda_{\rm s}$CDM retains its success in fitting the data and
addressing major cosmological tensions, without suggesting a need for a
deviation from the standard model of particle physics, and (ii) to determine
whether the data indicate new physics that could potentially address
cosmological tensions, either in the post-recombination universe through the
late-time mirror AdS-dS transition, or in the pre-recombination universe
through modifications in the standard values of $N_{\rm eff}$ and $\sum m_\nu$,
or both. Within the extended $\Lambda_{\rm s}$CDM model, referred to as
$\Lambda_{\rm s}$CDM+$N_{\rm eff}$+$\sum m_{\rm \nu}$, we find no significant
tension when considering the Planck-alone analysis. We observe that
incorporating BAO data limits the further success of the $\Lambda_{\rm s}$CDM
extension. However, the weakly model-dependent BAOtr data, along with Planck
and Planck+PP\&SH0ES, favor $H_0\sim 73\,{\rm km\, s^{-1}\, Mpc^{-1}}$. In
cases where BAOtr dataset is used, the mirror AdS-dS transition is very
effective in providing enhanced $H_0$ values, and thus the model requires no
significant deviation from the standard value of $N_{\rm eff} = 3.044$. Both
the $H_0$ and $S_8$ tensions are effectively addressed, with some compromise in
the case of the Planck+BAO dataset. Finally, the upper bounds obtained on $\sum
m_\nu \lesssim 0.5$~eV are fully compatible with neutrino oscillation
experiments. Our findings provide evidence that late-time physics beyond
$\Lambda$CDM, such as $\Lambda_{\rm s}$CDM, without altering the standard
pre-recombination universe, can suffice to alleviate the major cosmological
tensions.
