  The majoron, a neutrinophilic pseudo-Goldstone boson conventionally arising
in the context of neutrino mass models, can damp neutrino free-streaming and
inject additional energy density into neutrinos prior to recombination. The
combination of these effects for an eV-scale mass majoron has been shown to
ameliorate the outstanding $H_0$ tension, however only if one introduces
additional dark radiation at the level of $\Delta N_{\rm eff} \sim 0.5$. We
show here that models of low-scale leptogenesis can naturally source this dark
radiation by generating a primordial population of majorons from the decays of
GeV-scale sterile neutrinos in the early Universe. Using a posterior predictive
distribution conditioned on Planck2018+BAO data, we show that the value of
$H_0$ observed by the SH$_0$ES collaboration is expected to occur at the level
of $\sim 10\%$ in the primordial majoron cosmology (to be compared with $\sim
0.1\%$ in the case of $\Lambda$CDM). This insight provides an intriguing
connection between the neutrino mass mechanism, the baryon asymmetry of the
Universe, and the discrepant measurements of $H_0$.
