The mismatch between the locally measured expansion rate of the universe and the one inferred from observations of the cosmic microwave background (CMB) assuming the canonical $Lambda$CDM model has become the new cornerstone of modern cosmology, and many new-physics set ups are rising to the challenge. Concomitant with the so-called $H_0$ problem, there is evidence of a growing tension between the CMB-preferred value and the local determination of the weighted amplitude of matter fluctuations $S_8$. It would be appealing and compelling if both the $H_0$ and $S_8$ tensions were resolved at once, but as yet none of the proposed new-physics models have done so to a satisfactory degree. Herein, we adopt a systematic approach to investigate the possible interconnection among the free parameters in several classes of models that typify the main theoretical frameworks tackling the tensions on the universe expansion rate and the clustering of matter. Our calculations are carried out using the publicly available Boltzmann solver CAMB in combination with the sampler CosmoMC. We show that even after combining the leading classes of models sampling modifications of both the early and late time universe a simultaneous solution to the $H_0$ and $S_8$ tensions remains elusive.
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