  Observations of $z \sim 6$ quasars powered by supermassive black holes
(SMBHs; $M_{\rm BH} \sim 10^{8-10}\, M_\odot$) challenge our current
understanding of early black hole (BH) formation and evolution. The advent of
the James Webb Space Telescope (JWST) has enabled the study of massive BHs
(MBHs; $M_{\rm BH}\sim 10^{6-7} \ \mathrm{M}_\odot$) up to $z\sim 11$, thus
bridging the properties of $z\sim 6$ quasars to their ancestors. The JWST
spectroscopic observations of GN-z11, a well-known $z=10.6$ star-forming
galaxy, have been interpreted with the presence of a super-Eddington (Eddington
ratio $\equiv \,\lambda_{\rm Edd}\sim 5.5$) accreting MBH. To test this
hypothesis, we used a zoom-in cosmological simulation of galaxy formation and
BH co-evolution. We first tested the simulation results against the observed
probability distribution function (PDF) of $\lambda_{\rm Edd}$ found in $z\sim
6$ quasars. Then, in the simulation we selected the BHs that satisfy the
following criteria: (a) $10 < z < 11 $, (b) $M_{\rm BH} > 10^6 \
\mathrm{M}_\odot$. Next, we apply the extreme value statistics to the PDF of
$\lambda_{\rm Edd}$ resulting from the simulation and we find that the
probability of observing a $z\sim 10-11$ MBH accreting with $\lambda_{\rm Edd}
\sim 5.5$ in the volume surveyed by JWST is very low ($<0.2\%$). We compared
our predictions with those in the literature, and discuss the main limitations
of our work. Our simulation cannot explain the JWST observations of GN-z11.
This might be due to (i) poor resolution and statistics in simulations, (ii)
simplistic sub-grid models (e.g. BH accretion and seeding), (iii) uncertainties
in the data analysis and interpretation.
