  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 $w(z)$.
