  Recent outcomes by the DESI Collaboration have shed light on a possible
slightly evolving dark energy, challenging the standard $\Lambda$CDM paradigm.
To better understand dark energy nature, high-redshift observations like
gamma-ray burst data become essential for mapping the universe expansion
history, provided they are calibrated with other probes. To this aim, we
calibrate the $E_p-E_{iso}$ (or Amati) correlation through model-independent
B\'ezier interpolations of the updated Hubble rate and the novel DESI data
sets. More precisely, we provide two B\'ezier calibrations: i) handling the
entire DESI sample, and ii) excluding the point at $z_{eff}=0.51$, criticized
by the recent literature. In both the two options, we let the comoving sound
horizon at the drag epoch, $r_d$, vary in the range $r_d \in [138, 156]$ Mpc.
The Planck value is also explored for comparison. By means of the so-calibrated
gamma-ray bursts, we thus constrain three dark energy frameworks, namely the
standard $\Lambda$CDM, the $\omega_0$CDM and the $\omega_0\omega_1$CDM models,
in both spatially flat and non-flat universes. To do so, we worked out Monte
Carlo Markov chain analyses, making use of the Metropolis-Hastings algorithm.
Further, we adopt model selection criteria to check the statistically preferred
cosmological model finding a preference towards the concordance paradigm only
whether the spatial curvature is zero. Conversely, and quite interestingly, the
flat $\omega_0$CDM and both the cases, flat/non-flat, $\omega_0\omega_1$CDM
model, leave evidently open the chance that dark energy evolves at higher
redshifts.
