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)
