  We present an analysis aimed at combining cosmological constraints from
number counts of galaxy clusters identified through the Sunyaev-Zeldovich
effect, obtained with the South Pole Telescope (SPT), and from Lyman-$\alpha$
spectra obtained with the MIKE/HIRES and X-shooter spectrographs. The SPT
cluster analysis relies on mass calibration based on weak lensing measurements,
while the Lyman-$\alpha$ analysis is built over a suite of hydrodynamical
simulations for the extraction of mock spectra. The resulting constraints
exhibit a tension ($\sim 3.3\sigma$) between the low $\sigma_8$ values
preferred by the low-redshift cluster data, $\sigma_8=0.74 ^{+0.03}_{-0.04}$,
and the higher one preferred by the high-redshift Lyman-$\alpha$ data,
$\sigma_8=0.91 ^{+0.03}_{-0.03}$. We present a detailed analysis in order to
understand the origin of this tension and, in particular, to establish whether
it arises from systematic uncertainties related to the assumptions underlying
the analyses of cluster counts and/or Lyman-$\alpha$ forest. We found this
tension to be robust with respect to the choice of modeling of the IGM, even
when including possible systematics from unaccounted sub-Damped Lyman-$\alpha$
(DLA) and Lyman-limit systems (LLS) in the Lyman-$\alpha$ data. We conclude
that to solve this tension from the SPT side would require a large bias on the
cluster mass estimate, or from the Lyman-$\alpha$ side large unaccounted errors
on the Lyman-$\alpha$ mean fluxes, respectively. Our results have important
implications for future analyses based on cluster number counts from future
large photometric surveys (e.g. Euclid and LSST) and on larger samples of
high-redshift quasar spectra (e.g. DESI and WEAVE surveys). If confirmed at the
much higher statistical significance reachable by such surveys, this tension
could represent a significant challenge for the standard $\Lambda$CDM paradigm.
