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We present measurements of the size of the quasar proximity zone ($R_p$) for eleven low-luminosity ($-26.16\leq M_{1450}\leq-22.83$) quasars at $z\sim6$, discovered by the Subaru High-$z$ Exploration of Low-Luminosity Quasars project (SHELLQs). Our faint quasar sample expands the $R_p$ measurement down to $M_{1450}=-22.83$ mag, where more common quasar populations dominate at the epoch. We restrict the sample to quasars whose systemic redshifts have been precisely measured by [CII] 158 $μ$m or MgII $λ$2798 emission lines. We also update the $R_p$ measurements for 26 luminous quasars presented in Eilers et al. (2017)(arXiv:1703.02539) by using the latest systemic redshift results. The luminosity dependence on $R_p$ is found to be consistent with the theoretical prediction assuming highly ionized intergalactic medium. We find a shallow redshift evolution of the luminosity-corrected $R_p$, $R_{p,{\rm corr}}^{-25}$ ($R_{p, \rm corr}^{-25}\propto(1+z)^{-3.79\pm1.72}$) over $5.8\lesssim z \lesssim6.6$. This trend is steeper than that of Eilers et al. (2017) but significantly shallower than those of the earlier studies. Our results suggest that $R_{p,\rm corr}$ is insensitive to the neutral fraction of the universe at $z\sim6$. Four quasars show exceptionally small $R_{p,\rm corr}^{-25}$ ($ \lesssim0.90$ proper Mpc), which could be the result of their young age ($<10^4$ yr) in the reionization epoch, though statistics is still small.