  Cosmological models and their parameters are widely debated, especially about
whether the current discrepancy between the values of the Hubble constant,
$H_{0}$, obtained by type Ia supernovae (SNe Ia), and the Planck data from the
Cosmic Microwave Background Radiation could be alleviated when alternative
cosmological models are considered. Thus, combining high-redshift probes, such
as Gamma-Ray Bursts (GRBs) and Quasars (QSOs), together with Baryon Acoustic
Oscillations (BAO) and SNe Ia is important to assess the viability of these
alternative models and if they can cast further light on the Hubble tension. In
this work, for GRBs, we use a 3-dimensional relation between the peak prompt
luminosity, the rest-frame time at the end of the X-ray plateau, and its
corresponding luminosity in X-rays: the 3D Dainotti fundamental plane relation.
Regarding QSOs, we use the Risaliti-Lusso relation among the UV and X-ray
luminosities for a sample of 2421 sources. We correct both the QSO and GRB
relations by accounting for selection and evolutionary effects with a reliable
statistical method. We here use both the traditional Gaussian likelihoods
($\cal L_G$) and the new best-fit likelihoods ($\cal L_N$) to infer
cosmological parameters of a non-flat $\Lambda$CDM and flat $w$CDM models. We
obtain for all the parameters reduced uncertainties, up to $35\%$ for $H_{0}$,
when applying the new $\cal L_N$ likelihoods in place of the Gaussian ones. Our
results remain consistent with a flat $\Lambda$CDM model, although with a shift
of the dark energy parameter $w$ toward $w<-1$ and a curvature density
parameter toward $\Omega_k<0$.
