  We investigate the black hole mass function at $z\sim5$ using XQz5, our
recent sample of the most luminous quasars between the redshifts $4.5 < z <
5.3$. We include 72 quasars with black hole masses estimated from
velocity-broadened emission-line measurements and single-epoch virial
prescriptions in the footprint of a highly complete parent survey. The sample
mean Eddington ratio and standard deviation is $\log\lambda \approx
-0.20\pm0.24$. The completeness-corrected mass function is modelled as a double
power-law, and we constrain its evolution across redshift assuming
accretion-dominated mass growth. We estimate the evolution of the mass function
from $z=5-4$, presenting joint constraints on accretion properties through a
measured dimensionless e-folding parameter, $k_{\rm{ef}} \equiv
\langle\lambda\rangle U (1-\epsilon)/\epsilon = 1.79\pm0.06$, where
$\langle\lambda\rangle$ is the mean Eddington ratio, $U$ is the duty cycle, and
$\epsilon$ is the radiative efficiency. If these supermassive black holes were
to form from seeds smaller than $10^8\,M_{\odot}$, the growth rate must have
been considerably faster at $z\gg5$ than observed from $z=5-4$. A growth rate
exceeding $3\times$ the observed rate would reduce the initial heavy seed mass
to $10^{5-6}\,M_{\odot}$, aligning with supermassive star and/or direct
collapse seed masses. Stellar mass ($10^2\,M_{\odot}$) black hole seeds would
require $\gtrsim4.5\times$ the observed growth rate at $z\gg5$ to reproduce the
measured active black hole mass function. A possible pathway to produce the
most extreme quasars is radiatively inefficient accretion flow, suggesting
black holes with low angular momentum or photon trapping in supercritically
accreting thick discs.
