  Long gamma-ray burst (GRB) prompt emission shows a correlation between the
intrinsic peak energy, $E_{\mathrm{p,i}}$, of the time-average $\nu F_{\nu}$
spectrum and the isotropic-equivalent peak gamma-ray luminosity, $L_{{\rm
p,iso}}$, as well as the total released energy, $E_{\rm iso}$. The same
correlation is found within individual bursts, when time-resolved $E_{\rm p,i}$
and $L_{\rm iso}$ are considered. These correlations are characterised by an
intrinsic dispersion, whose origin is still unknown. Discovering the origin of
the correlation and of its dispersion would shed light on the still poorly
understood prompt emission and would propel GRBs to powerful standard candles.
We studied the dispersion of both isotropic-equivalent and
collimation-corrected time-resolved correlations. We also investigated whether
the intrinsic dispersion computed within individual GRBs is different from that
obtained including different bursts into a unique sample. We then searched for
correlations between key features, like Lorentz factor and jet opening angle,
and intrinsic dispersion, when the latter is treated as one of the
characterising We performed a time-resolved spectral analysis of 20 long
Type-II or collapsar-candidate GRBs detected by the Fermi Gamma-ray Burst
Monitor with known redshift and estimates of jet opening angle and/or Lorentz
factor. The collimation-corrected correlation appears to be no less dispersed
than the isotropic-equivalent one. Also, individual GRBs are significantly less
dispersed than the whole sample. We excluded (at $4.2 \sigma$ confidence level)
the difference in samples' sizes as the possible reason, thus confirming that
individual GRBs are {\em intrinsically} less dispersed than the whole sample.
No correlation was found between intrinsic dispersion and other key properties
for the few GRBs with available information.
