Cluster counts III. $\Lambda$CDM extensions and the cluster tension

In this work, we examine whether further extensions to the $\Lambda$CDM model could alleviate the $\sigma_8$ tension. For that, we derive constraints on the parameters subject of the discrepancy, using CMB $C_{\ell}$ combined with cluster counts SZ sample with a free dark energy equation of state parameter while allowing the clusters mass calibration parameter $(1-b)$ to vary. The latter is degenerate with $\sigma_8$, which translates the discrepancy within $\Lambda$CDM framework into one between $(1-b) \sim0.6$, corresponding to constraints on $\sigma_8$ obtained from CMB, and $(1-b)\sim0.8$, the value adopted for the SZ sample calibration. We find that a constant $w$, when left free to vary along with large priors on the matter density ([0.1,1.0]) and Hubble parameters ([30,200]), can reduce the discrepancy to less than 2$\sigma$ for values far below its fiducial $w$ = -1. However, the latter were not allowed when we additionally combine with other probes like BAO feature angular diameter distance measured in galaxies clustering surveys. We found also, when we allow to vary, in addition to $w$, a modification of the growth rate through the growth index $\gamma$, that the tension is alleviated, with $(1-b)$ likelihood now centered around the Planck calibration value of $\sim$ 0.8. However, here again, combining with geometrical distance probes restores the discrepancy, with $(1-b)$ preferred value reverting back to the $\Lambda$CDM one of $\sim$ 0.6. The same situation is observed when introducing, along with $w$ and $\gamma$, further extensions to $\Lambda$CDM like massive neutrinos, although the latter allows to reduce the tension to 2$\sigma$, even when combining with BAO datasets. We conclude that none of these common extensions to $\Lambda$CDM is able to fix the discrepancy and a misdetermination of the calibration factor is the most preferred explanation... (Abridged)