Testing the hypothesis of a matter density discrepancy within LCDM model using multiple probes

We investigate whether the two cosmological discrepancies on the Hubble constant ($H_0$) and the matter fluctuation parameter ($\sigma_8$) are suggesting and compatible with the existence of an additional one on the matter density. Knowing that the latter effects on observables is degenerate with those coming from $H_0$ and $\sigma_8$, we combined different probes to break these degeneracies while adopting the agnostic approach of, either relaxing the calibration parameters in each probe, or by only including priors with the condition that they are obtained independently from the discrepant parameters. We also compiled and used a dataset from previous direct measurements of $\Omega_{\rm{M}}$ obtained in a model independent way using the Oort technique. We found when combining galaxy cluster counts + cluster gas mass fraction probe + cosmic chronometers + direct $\Omega_{\rm{M}}$ + priors from BBN and CMB, that both parameters, $H_0$ and $\sigma_8$, are consistent with those inferred from local probes, with $\sigma_8 = 0.745 \pm 0.05$ while $H_0 = 73.8 \pm 3.01$, and that for a value of $\Omega_{\rm{M}} = 0.22 \pm 0.01$ at more than 3$\sigma$ from that determined by the CMB. However discrepancies appeared when we combined SN in addition to CC suggesting either inconsistencies between the SN sample and the other probes or a challenge to our hypothesis, while only a prior on the matter density obtained from the CMB data keeps $\sigma_8$ within the values usually obtained when adopting the calibration parameters of the low redshift growth of structures probes. We conclude that, either both tensions are compatible with the local inferred low values of matter density at odd with those obtained by CMB, reviving by then an overlooked discrepancy, or that the $\Lambda$CDM model is facing more difficulties to accommodate simultaneously all the current available observations.(abridged)