  With the recent increase in precision of our cosmological datasets,
measurements of $\Lambda$CDM model parameter provided by high- and low-redshift
observations started to be in tension, i.e., the obtained values of such
parameters were shown to be significantly different in a statistical sense.
In~this work we tackle the tension on the value of the Hubble parameter, $H_0$,
and the weighted amplitude of matter fluctuations, $S_8$, obtained from local
or low-redshift measurements and from cosmic microwave background (CMB)
observations. We combine the main approaches previously used in the literature
by extending the cosmological model and accounting for extra systematic
uncertainties. With such analysis we aim at exploring non standard cosmological
models, implying deviation from a cosmological constant driven acceleration of
the Universe expansion, in the presence of additional uncertainties in
measurements. In more detail, we reconstruct the Dark Energy equation of state
as a function of redshift, while we study the impact of type-Ia supernovae
(SNIa) redshift-dependent astrophysical systematic effects on these tensions.
We consider a SNIa intrinsic luminosity dependence on redshift due to the star
formation rate in its environment, or the metallicity of the progenitor. We
find that the $H_0$ and $S_8$ tensions can be significantly alleviated, or~even
removed, if we account for varying Dark Energy for SNIa and CMB data. However,
the tensions remain when we add baryon acoustic oscillations (BAO) data into
the analysis, even after the addition of extra SNIa systematic uncertainties.
This points towards the need of either new physics beyond late-time Dark
Energy, or other unaccounted systematic effects (particulary in BAO
measurements), to fully solve the present tensions.
