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 $sum m_ u$ 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 $gamma=0.60pm 0.13$. 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 $2sigma$ level) provided $sum m_ usim0.34 $ eV and $gammasim0.8$. 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.
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