  The mismatch between the locally measured expansion rate of the universe and
the one inferred from the cosmic microwave background measurements by Planck in
the context of the standard $\Lambda$CDM, known as the Hubble tension, has
become one of the most pressing problems in cosmology. A large number of
amendments to the $\Lambda$CDM model have been proposed in order to solve this
tension. Many of them introduce new physics, such as early dark energy,
modifications of the standard model neutrino sector, extra radiation,
primordial magnetic fields or varying fundamental constants, with the aim of
reducing the sound horizon at recombination $r_{\star}$. We demonstrate here
that any model which only reduces $r_{\star}$ can never fully resolve the
Hubble tension while remaining consistent with other cosmological datasets. We
show explicitly that models which achieve a higher Hubble constant with lower
values of matter density $\Omega_m h^2$ run into tension with the observations
of baryon acoustic oscillations, while models with larger $\Omega_mh^2$ develop
tension with galaxy weak lensing data.
