Some cosmological models with non-negligible dark energy fractions in
particular windows of the pre-recombination epoch are capable of alleviating
the Hubble tension quite efficiently, while keeping the good description of the
data that are used to build the cosmic inverse distance ladder. There has been
an intensive discussion in the community on whether these models enhance the
power of matter fluctuations, leading {\it de facto} to a worsening of the
tension with the large-scale structure measurements. We address this pivotal
question in the context of several early dark energy (EDE) models, considering
also in some cases a coupling between dark energy and dark matter, and the
effect of massive neutrinos. We fit them using the Planck 2018 likelihoods, the
supernovae of Type Ia from the Pantheon compilation and data on baryon acoustic
oscillations. We find that ultra-light axion-like (ULA) EDE can actually
alleviate the $H_0$ tension without increasing the values of $\sigma_{12}$ with
respect to those found in the $\Lambda$CDM, whereas EDE with an exponential
potential does not have any impact on the tensions. A coupling in the dark
sector tends to enhance the clustering of matter, and the data limit a lot the
influence of massive neutrinos, since the upper bounds on the sum of their
masses are too close to those obtained in the standard model. We find that in
the best case, namely ULA, the Hubble tension is reduced to $\sim 2\sigma$.
