  The recent $\sim 4 \, \sigma$ Hubble constant, $H_{0}$, tension is observed
between the value of $H_{0}$ from the Cosmic Microwave Background (CMB) and
Type Ia Supernovae (SNe Ia). It is a decade since this tension is excruciating
the modern astrophysical community. To shed light on this problem is key to
consider probes at intermediate redshifts between SNe Ia and CMB and reduce the
uncertainty on $H_0$. Toward these goals, we fill the redshift gap by employing
Gamma-Ray Bursts (GRBs) and Quasars (QSOs), reaching $z=9.4$ and $z=7.6$,
respectively, combined with Baryonic Acoustic Oscillations (BAO) and SNe Ia. To
this end, we employ the ``Dainotti GRB 3D relation" among the rest-frame end
time of the X-ray plateau emission, its corresponding luminosity, and the peak
prompt luminosity, and the ``Risaliti-Lusso" QSO relation between ultraviolet
and X-ray luminosities. We inquire the commonly adopted Gaussianity assumption
on GRBs, QSOs, and BAO. With the joint sample, we fit the flat $\Lambda$ Cold
Dark Matter model with both the Gaussian and the newly discovered likelihoods.
We also investigate the impact of the calibration assumed for \textit{Pantheon}
and \textit{Pantheon +} SNe Ia on this analysis. Remarkably, we show that only
GRBs fulfill the Gaussianity assumption. We achieve small uncertainties on the
matter density parameter $\Omega_M$ and $H_0$. We find $H_0$ values compatible
within 2 $\sigma$ with the one from the Tip of the Red Giant Branch. Finally,
we show that the cosmological results are heavily biased against the arbitrary
calibration choice for SNe Ia.
