  The Hubble tension can be addressed by modifying the sound horizon ($r_s$)
before recombination, triggering interest in $r_s$-free early-universe
estimates of the Hubble constant, $H_0$. Constraints on $H_0$ from an
$r_s$-free analysis of the full shape BOSS galaxy power spectra within LCDM
were recently reported and used to comment on the viability of physics beyond
LCDM. Here we demonstrate that $r_s$-free analyses with current data depend on
the model and the priors placed on the cosmological parameters, such that LCDM
analyses cannot be used as evidence for or against new physics. We find that
beyond-LCDM models which introduce additional energy density with significant
pressure support, such as early dark energy (EDE) or additional neutrino energy
density ($\Delta N_{\rm eff}$), lead to larger values of $H_0$. On the other
hand, models which only affect the time of recombination, such as a varying
electron mass ($\Delta m_e$), produce $H_0$ constraints similar to LCDM. Using
BOSS data, constraints from light element abundances, cosmic microwave
background (CMB) lensing, a CMB-based prior on the scalar amplitude ($A_s$),
spectral index ($n_s$), and $\Omega_m$ from the Pantheon+ supernovae data set,
we find that in LCDM, $H_0=64.9\pm 2.2$ km/s/Mpc; EDE, $H_0=68.7^{+3}_{-3.9}$;
$\Delta N_{\rm eff}$, $H_0=68.1^{+2.7}_{-3.8}$; $\Delta m_e$,
$H_0=64.7^{+1.9}_{-2.3}$. Using a prior on $\Omega_m$ from uncalibrated BAO and
CMB measurements of the projected sound horizon, these values become in LCDM,
$H_0=68.8^{+1.8}_{-2.1}$; EDE, $H_0=73.7^{+3.2}_{-3.9}$; $\Delta N_{\rm eff}$,
$H_0=72.6^{+2.8}_{-3.7}$; $\Delta m_e$, $H_0=68.8\pm 1.9$. With current data,
none of the models are in significant tension with SH0ES, and consistency tests
based on comparing $H_0$ posteriors with and without $r_s$ marginalization are
inconclusive with respect to the viability of beyond LCDM models.
