for our potential $z\simeq6$ companion galaxies, with a mean of $n=4.3$, this correlation may explain, at least in part, our larger measured sizes.\
Using the number of galaxies observed in HST data of comparable depth [WFC3 ERS2 field, $m_H<26.5$; @Windhorst2011 see their figure 12], the average number of galaxies observed is $\approx373,000$ per square degree, or 0.0288 objects per square arcseconds, [*if*]{} galaxies are uniformly distributed on the sky. We thus expect to find on average 7.3 [*random*]{} foreground objects within our six $\approx 6\farcs5 \times 6\farcs5$ quasar images (total area $\approx250$ square arcseconds; Figures \[fig:quasar1\_resids\]–\[fig:quasar3\_resids\]), at $z\ll6$ and unrelated to our quasars. The surface density variations in these numbers due to foreground cosmic variance and photometric zeropoint errors is expected to be $\lesssim$15% in the H-band [see, e.g., figure 3 of @Driver2016], so we would on average expect $\lesssim$ 8.4 random foreground objects at $z\ll6$.

Assuming that the galaxy neighbor distribution follows a Poisson distribution, the probability of observing a total of 20 galaxies within the $\approx250$ square arcsecond area, $4\sigma$ above the expected value of 8.4 foreground galaxies, is $\lesssim0.0003$. Hence, given the unlikelihood that so many galaxies would be found by chance, it is probable that some 