  The $\Lambda$CDM concordance model is very successful at describing our
Universe with high accuracy and few parameters. Despite its successes, a few
tensions persist; most notably, the best-fit $\Lambda$CDM model, as derived
from the Planck CMB data, largely overpredicts the abundance of SZ clusters
when using their standard mass calibration. Whether this is a sign of an
incorrect calibration or the need for new physics remains a matter of debate.
Here we examined two simple extensions of the standard model and their ability
to release this tension: massive neutrinos and a simple modified gravity model
via a non-standard growth index $\gamma$. We used both the Planck CMB and SZ
cluster counts as datasets, with or without local X-ray clusters. In the case
of massive neutrinos, the SZ calibration $(1-b)$ is constrained to
$0.59^{+0.03}_{-0.04}$ (68\%), more than 5$\sigma$ away from its standard value
$\sim0.8$. We found little correlation between $\sum m_\nu$ and $(1-b)$,
corroborating previous conclusions derived from X-ray clusters; massive
neutrinos do not alleviate the cluster-CMB tension. With our simple $\gamma$
model, we found a large correlation between calibration and growth index but
contrary to local X-ray clusters, SZ clusters are able to break the degeneracy
between the two thanks to their extended $z$ range. The calibration $(1-b)$ was
then constrained to $0.60^{+0.05}_{-0.07}$, leading to an interesting
constraint on $\gamma=0.60\pm 0.13$. When both massive neutrinos and modified
gravity were allowed, preferred values remained centred on standard
$\Lambda$CDM values, but $(1-b)\sim0.8$ was allowed (though only at the
$2\sigma$ level) provided $\sum m_\nu\sim0.34 $ eV and $\gamma\sim0.8$. We
conclude that massive neutrinos do not relieve the cluster-CMB tension and that
a calibration close to the standard value $0.8$ would call for new physics in
the gravitational sector.
