COOL-LAMPS VIII: Known wide-separation lensed quasars and their host galaxies reveal a lack of evolution in $M_{\rm{BH}}/M_\star$ since $z\sim 3$
Wide-separation lensed quasars (WSLQs) are a rare class of strongly lensed quasars, magnified by foreground massive galaxy clusters, with typically large magnifications of the multiple quasar images. They are a relatively unexplored opportunity for detailed study of quasar host galaxies. The current small sample of known WSLQs has a median redshift of $z\approx 2.1$, larger than most other samples of quasar host galaxies studied to date. Here, we derive precise constraints on the properties of six WSLQs and their host galaxies, using parametric surface brightness fitting, measurements of quasar emission lines, and stellar population synthesis of host galaxies in six WSLQ systems. Our results, with significant uncertainty, indicate that these six hosts are a mixture of star-forming and quiescent galaxies. To probe for co-evolution between AGNs and host galaxies, we model the offset from the `local' ($z=0$) $M_{\rm{BH}}\unicode{x2013}M_\star$ relation as a simple power-law in redshift. Accounting for selection effects, a WSLQ-based model for evolution in the $M_{\rm{BH}}\unicode{x2013}M_\star$ relation has a power-law index of $\gamma_M=-0.42\pm0.31$, consistent with no evolution. Compared to several literature samples, which mostly probe unlensed quasars at $z<2$, the WSLQ sample shows less evolution from the local relation, at $\sim 4\sigma$. We find that selection affects and choices of $M_{\rm{BH}}$ calibration are the most important systematics in these comparisons. Given that we resolve host galaxy flux confidently even from the ground in some instances, our work demonstrates that WSLQs and highly magnified AGNs are exceptional systems for future AGN$\unicode{x2013}$host co-evolution studies.