  The Hubble tension is inherently multidimensional, and bears important
implications for parameters beyond $H_0$. We discuss the key role of the matter
density parameter $\Omega_m$ and the physical cold dark matter density
$\omega_c$. We argue that once $\Omega_m$ and the physical baryon density
$\omega_b$ are calibrated, through Baryon Acoustic Oscillations (BAO) and/or
Type Ia Supernovae (SNeIa) for $\Omega_m$, and via Big Bang Nucleosynthesis for
$\omega_b$, any model raising $H_0$ inevitably raises $\omega_c$ and, under
minimal assumptions, the clustering parameter $S_8$. We explicitly verify that
this behaviour holds when analyzing recent BAO and SNeIa data. We argue that a
calibration of $\Omega_m$ which is as reliable and model-independent as
possible should be a priority in the Hubble tension discussion, and an
interesting possibility in this sense could be represented by galaxy cluster
gas mass fraction measurements.
