  Enhanced ionizing radiation in close proximity to redshift $z\gtrsim 6$
quasars creates short windows of intergalactic Ly$\alpha$ transmission blueward
of the quasar Ly$\alpha$ emission lines. The majority of these Ly$\alpha$
near-zones are consistent with quasars that have optically/UV bright lifetimes
of $t_{\rm Q}\sim 10^{5}-10^{7}\rm\,yr$. However, lifetimes as short as $t_{\rm
Q}\lesssim 10^{4}\rm\,yr$ appear to be required by the smallest Ly$\alpha$
near-zones. These short lifetimes present an apparent challenge for the growth
of $\sim 10^{9}\rm\,M_{\odot}$ black holes at $z\gtrsim 6$. Accretion over
longer timescales is only possible if black holes grow primarily in an obscured
phase, or if the quasars are variable on timescales comparable to the
equilibriation time for ionized hydrogen. Distinguishing between very young
quasars and older quasars that have experienced episodic accretion with
Ly$\alpha$ absorption alone is challenging, however. We therefore predict the
signature of proximate 21-cm absorption around $z\gtrsim 6$ radio-loud quasars.
For modest pre-heating of intergalactic hydrogen by the X-ray background, where
the spin temperature $T_{\rm S} \lesssim 10^{2}\rm\,K$ prior to any quasar
heating, we find proximate 21-cm absorption should be observable in the spectra
of radio-loud quasars. The extent of the proximate 21-cm absorption is
sensitive to the integrated lifetime of the quasar. Evidence for proximate
21-cm absorption from the diffuse intergalactic medium within $2-3\rm\,pMpc$ of
a (radio-loud) quasar would be consistent with a short quasar lifetime, $t_{\rm
Q}\lesssim 10^{5}\rm\,yr$, and would provide a complementary constraint on
models for high redshift black hole growth.
