Can early black holes sustain accretion above the Eddington limit?
In plain words
A black hole normally cannot swallow gas faster than a limit set by the outward push of the light the infalling gas emits (the Eddington limit). Growing a billion-solar-mass black hole within 700 million years from a small seed seems to need this limit to be exceeded for a long time.
Precise statement
Determine whether seeds of $10^{2}-10^{5}\,M_{\mathrm{sun}}$ formed at $z \sim 20-30$ can sustain a duty-cycle-averaged Eddington ratio $\mathrm{mdot}/\mathrm{mdot}_{\mathrm{Edd}} \ge 1$, with $\mathrm{mdot}_{\mathrm{Edd}} = L_{\mathrm{Edd}}/(\eta c^2)$, $L_{\mathrm{Edd}} = 4 \pi G M m_p c / \sigma_T = 1.26x10^{38} (M/M_{\mathrm{sun}})\,\mathrm{erg/s}$ and $\eta$ the radiative efficiency, over the $\sim 5x10^{8}\,\mathrm{yr}$ available before $z \sim 7.5$ (e-folding time $\sim 4.5x10^{7} (\eta/0.1)\,\mathrm{yr}$), or must grow through super-Eddington episodes, despite radiative feedback, angular-momentum barriers and supernova-driven gas removal. The answer is yes or no with the required duty cycle, Eddington ratio and $\eta$, or the Eddington-ratio distribution that reproduces the $z > 6$ black hole mass function.
What would settle it
Radiation-hydrodynamic simulations resolving scales from inside the Bondi radius to the host galaxy, checked against measured Eddington-ratio distributions of $z > 6$ AGN.