  The $\Lambda$CDM model faces several tensions with recent cosmological data
and their increased accuracy. The mismatch between the values of the Hubble
constant $H_0$ obtained from direct distance ladder measurements and from the
cosmic microwave background (CMB) is the most statistically significant, but
the amplitude of the matter fluctuations is also regarded as a serious concern,
leading to the investigation of a plethora of models. We first show that the
combination of several recent measurements from local probes leads to a tight
constraint on the present-day matter density $\Omega_M$ as well as on the
amplitude of the matter fluctuations, both acceptably consistent with the
values inferred from the CMB. Secondly, we show that the data on cosmic
chronometers allow to derive an accurate value of the Hubble constant $H_0$ for
$\Lambda$CDM models: $H_0 = 67.4 \pm 1.34$ km/s/Mpc. This implies that, within
$\Lambda$CDM, some determinations of $H_0$ are biased. Considering a bias on
the Hubble constant as a nuisance parameter within $\Lambda$CDM, we examine
such a $\Lambda$CDM$+ H_0$ bias model on the same statistical grounds as
alternative cosmological models. We show that the former statistically
supersede most existing extended models proposed up to now. In a third step, we
show that the value of $\Omega_M$ we obtained, combined with $H_0$ from SH0ES,
leads to an accurate measurement of $\omega_M$, providing an additional
low-redshift test for cosmological models. From this test, most extensions seem
to be confronted with a new tension, whereas the $\Lambda$CDM with $H_0 \sim 67
$ has none. We conclude that a standard $\Lambda$CDM model with an unknown bias
in the Cepheids distance calibration represents a model that reaches a
remarkable agreement, statistically better than previously proposed extensions
with $H_0 \sim 73 $ for which such a comparison can be performed. (abridged)
