this contaminated inner region, we instead measure the surface brightness in annuli from 7 to 10 pixels, or 0$\farcs$42–0$\farcs$60. We choose an inner radius of 7 pixels, as this is where the pixel-to-pixel variations of the $S/N$ first reach the expected/background level. This approach ignores the central core, while including enough pixels to make a reasonable detection if any flux was present. For all quasars in both filters, no significant flux detection could be made in these annuli. The 2$\sigma$ surface brightness limits in these annuli, from the noise of each image, are given in Table \[tab:quasar\_photometry\].

To obtain the total magnitude limit of a given host, we consider a range of profiles, with a distribution of $n$ and $R_e$ guided by $z\simeq6$ observations [@Shibuya2015], and take a Monte Carlo approach to determine the most likely magnitude limit given the surface brightness limit in the annulus (see Appendix \[Appendix\]). The 2$\sigma$ magnitude limits obtained by this method are given in Table \[tab:quasar\_photometry\], with the J- and H-band limits ranging from 22.7–23.1 mag and 22.4–22.9 mag, respectively.

### Stellar Mass Limits {#StellarMassLimits}

Measuring the redshift evolution of the black hole–stellar mass relation is of key importance for understanding the co-evolution of 