  We perform a comprehensive analysis of the most common early- and
late-Universe solutions to the $H_0$, Ly-$\alpha$, and $S_8$ discrepancies.
When considered on their own, massive neutrinos provide a natural solution to
the $S_8$ discrepancy at the expense of increasing the $H_0$ tension. If all
extensions are considered simultaneously, the best-fit solution has a neutrino
mass sum of $\sim 0.4$ eV, a dark energy equation of state close to that of a
cosmological constant, and no additional relativistic degrees of freedom.
However, the $H_0$ tension, while weakened, remains unresolved. Motivated by
this result, we perform a non-parametric reconstruction of the evolution of the
dark energy fluid density (allowing for negative energy densities), together
with massive neutrinos. When all datasets are included, there exists a residual
$\sim1.9\sigma$ tension with $H_0$. If this residual tension remains in the
future, it will indicate that it is not possible to solve the $H_0$ tension
solely with a modification of the late-Universe dynamics within standard
general relativity. However, we do find that it is possible to resolve the
tension if either galaxy BAO or JLA supernovae data are omitted. We find that
\textit{negative} dark energy densities are favored near redshift $z\sim2.35$
when including the Ly-$\alpha$ BAO measurement (at $\sim 2\sigma$). This
behavior may point to a negative curvature, but it is most likely indicative of
systematics or at least an underestimated covariance matrix. Quite remarkably,
we find that in the extended cosmologies considered in this work, the neutrino
mass sum is always close to $0.4$ eV regardless of the choice of external
datasets, as long as the $H_0$ tension is solved or significantly decreased.
