A generic $\omega_b$ tension in early-time solutions to the Hubble tension

I show that early-time (pre-recombination) solutions to the Hubble tension are generically expected to increase the preferred baryon density $\omega_b$. This puts these models in tension with Big Bang Nucleosynthesis (BBN), as measurements of primordial deuterium constrain $\omega_b$ at percent level. I show that existing analyses are in tension with the BBN determination of $\omega_b$, and that including a likelihood component for primordial deuterium deters two representative models from recovering a high $H_0$, and leads to worse fits to CMB, BAO, supernova, and BBN data than $\Lambda$CDM.