  The strong discrepancy between local and inverse distance ladder estimates of
the Hubble constant $H_0$ could be pointing towards new physics beyond
$\Lambda$CDM. Several attempts to address this tension through new physics rely
on extended models, featuring extra free parameters beyond the 6 $\Lambda$CDM
parameters. However, marginalizing over extra parameters has the effect of
broadening the uncertainties on the inferred parameters, and it is often the
case that within these models the $H_0$ tension is addressed due to larger
uncertainties rather than a genuine shift in the central value of $H_0$. What
happens if a physical theory is able to fix the extra parameters to a specific
set of non-standard values? The degrees of freedom of the model are reduced
with respect to the standard case where the extra parameters are free to vary.
Focusing on the dark energy equation of state $w$ and the effective number of
relativistic species $N_{\rm eff}$, I find that physical theories able to fix
$w \approx -1.3$ or $N_{\rm eff} \approx 3.95$ would lead to an estimate of
$H_0$ from CMB, BAO, and SNeIa data in perfect agreement with the local
distance ladder estimate, without broadening its uncertainty. These two
non-standard models are, from a model-selection perspective, strongly
disfavoured with respect to $\Lambda$CDM. However, models that predict $N_{\rm
eff} \approx 3.45$ would be able to bring the tension down to $1.5\sigma$ while
only being weakly disfavored with respect to $\Lambda$CDM, whereas models that
predict $w \approx -1.1$ would be able to bring the tension down to $2\sigma$
(at the cost of the preference for $\Lambda$CDM being definite). Finally, I
estimate dimensionless multipliers relating variations in $H_0$ to variations
in $w$ and $N_{\rm eff}$, which can be used to repeat the analysis of this
paper in light of future more precise local distance ladder estimates of $H_0$.
