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We make the case that there can be no low-redshift solution to the $H_0$ tension. To robustly answer this question, we use a very flexible parameterization for the dark energy equation of state such that every cosmological distance still allowed by data exists within this prior volume. To then answer whether there exists a satisfactory solution to the $H_0$ tension within this comprehensive parameterization, we constrained the parametric form using different partitions of the Planck cosmic microwave background, SDSS-IV/eBOSS DR16 baryon acoustic oscillation, and Pantheon supernova datasets. When constrained by just the cosmic microwave background dataset, there exists a set of equations of state which yields high $H_0$ values, but these equations of state are ruled out by the combination of the supernova and baryon acoustic oscillation datasets. In other words, the constraint from the cosmic microwave background, baryon acoustic oscillation, and supernova datasets together does not allow for high $H_0$ values and converges around an equation of state consistent with a cosmological constant. Thus, since this very flexible parameterization does not offer a solution to the $H_0$ tension, there can be no solution to the $H_0$ tension that adds physics at only low redshifts. This is directly related to the expansion history of the Universe and its geometrical properties and would include models beyond those parametrized by $w(z)$.