  We study quasar proximity zones in the redshift range $5.77 \leq z \leq 6.54$
by homogeneously analyzing $34$ medium resolution spectra, encompassing both
archival and newly obtained data, and exploiting recently updated systemic
redshift and magnitude measurements. Whereas previous studies found strong
evolution of proximity zone sizes with redshift, and argued that this provides
evidence for a rapidly evolving intergalactic medium (IGM) neutral fraction
during reionization, we measure a much shallower trend $\propto(1+z)^{-1.44}$.
We compare our measured proximity zone sizes to predictions from hydrodynamical
simulations post-processed with one-dimensional radiative transfer, and find
good agreement between observations and theory irrespective of the ionization
state of the ambient IGM. This insensitivity to IGM ionization state has been
previously noted, and results from the fact that the definition of proximity
zone size as the first drop of the smoothed quasar spectrum below the $10\%$
flux transmission level probes locations where the ionizing radiation from the
quasar is an order of magnitude larger than the expected ultraviolet ionizing
background that sets the neutral fraction of the IGM. Our analysis also
uncovered three objects with exceptionally small proximity zones (two have $R_p
< 1$proper Mpc), which constitute outliers from the observed distribution and
are challenging to explain with our radiative transfer simulations. We consider
various explanations for their origin, such as strong absorption line systems
associated with the quasar or patchy reionization, but find that the most
compelling scenario is that these quasars have been shining for $\lesssim
10^5$yr.
