= Solution
Distinguish the rest-mass supply $\dot M_{\rm in}$ from the actual black-hole mass growth. With the usual radiative-efficiency definition,
$$
L=\epsilon\dot M_{\rm in}c^2,\qquad
\dot M_{\rm BH}=(1-\epsilon)\dot M_{\rm in}.
$$
At the Eddington limit the growth is exponential:
$$
\frac{\dot M_{\rm BH}}{M_{\rm BH}}
=\frac{1-\epsilon}{\epsilon}
\frac{4\pi Gm_p}{\sigma_e c}
=\frac1{t_{\rm Sal}},\qquad
t_{\rm Sal}=\frac{\epsilon}{1-\epsilon}
\frac{\sigma_e c}{4\pi Gm_p}.
$$
The available time is the difference of the supplied ages, \b[$\Delta t=650\ {\rm Myr}$]; no new cosmological age-redshift relation is needed.
Using the printed $\sigma_e=6.6\times10^{-29}\ {\rm m^2}$, together with $G=6.67430\times10^{-11}\ {\rm m^3\,kg^{-1}\,s^{-2}}$, $m_p\simeq1.67262\times10^{-27}\ {\rm kg}$, $c=299792458\ {\rm m\,s^{-1}}$ and a Julian year of $31557600\ {\rm s}$, gives
$$
t_E=\frac{\sigma_e c}{4\pi Gm_p}\simeq446.94\ {\rm Myr},
\qquad t_{\rm Sal}\simeq49.66\ {\rm Myr}.
$$
Therefore
$$
\boxed{M_{{\rm BH},0,\min}
=(2\times10^9M_\odot)e^{-650/49.66}
\simeq4.1\times10^3M_\odot.}
$$
This is about $13.09$ e-folds. The constants just listed are evaluation constants, not additional observational inputs asserted to be on the paper.
Sometimes an elementary growth convention identifies $\dot M_{\rm BH}$ directly with $L/(\epsilon c^2)$ and drops the radiated fraction of the incoming rest mass. That approximation instead gives $t_{\rm Sal}\simeq44.69\ {\rm Myr}$ and $M_0\simeq9.7\times10^2M_\odot$. The first result retains the physically distinct supplied and retained masses; the convention must be stated because exponential growth magnifies this difference.
The main implication is already severe in the optimistic calculation: a low-mass stellar-remnant seed has insufficient time under the specified efficiency and strictly Eddington-limited <accretion>. A heavier seed, earlier formation, a different efficiency, mergers or episodes beyond that idealized <luminosity> limit can relax the inference. Any <duty cycle> below unity or sub-Eddington episodes require a larger initial mass within the adopted model. The reported high-redshift quasar motivating this calculation was identified spectroscopically at $z=7.085$; its published age estimate need not replace the ages supplied here.<A luminous quasar at a redshift of z = 7.085 by Daniel J. Mortlock and collaborators>[2011 abstract, redshift and inferred black-hole mass]{ref}
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