We perform a comprehensive analysis of the most common early-and late-Universe solutions to the H0, Ly-α, and S8 discrepancies. When considered on their own, massive neutrinos provide a natural solution to the S8 discrepancy at the expense of increasing the H0 tension. If all extensions are considered simultaneously, the best-fit solution has a neutrino mass sum of ∼ 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 H0 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 ∼ 1.9σ tension with H0. If this residual tension remains in the future, it will indicate that it is not possible to solve the H0 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 negative dark energy densities are favored near redshift z ∼ 2.35 when including the Ly-α BAO measurement (at ∼ 2σ). 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 H0 tension is solved or significantly decreased.2 Another class of potential solutions involves interacting [14,20,21] or decaying dark matter [11][12][13] in an isolated dark sector.