.1456 & 349.63792 & -30.49266 &  2.3  $\pm$ 0.1  &  293  $\pm$ 17  & 6 & 39.9 & {\bf 0.1} & 1.7 & 2018.1.00908.S & uid://A001/X133d/X2619 \\
		\hline
		J2348--3054\textsuperscript{\ref{fn:uberres}} & 6.9007 & 357.13895 & -30.90285 &  1.5  $\pm$ 0.2  &  457  $\pm$ 49  & 1 & 16.7 & 0.41 & 5.3 & 2012.1.00882.S & uid://A002/X5a9a13/X53f \\
		&  &  &  &  & & 2\textsuperscript{\ref{fn:recal}}  & 36.8 & 0.19 & 2.0 & 2013.1.00273.S & uid://A001/X148/X73 \\
		&  &  &  &  & & 3 & 51.8 & {\bf 0.17} & 3.1 & 2015.1.00399.S & uid://A001/X5a3/X56 \\
		\hline
		P359--06 & 6.1719 & 359.13517 & -6.38313 &  2.7  $\pm$ 0.1  &  341  $\pm$ 18  & 3\textsuperscript{\ref{fn:recal}}  & 7.9 & 0.66 & 6.3 & 2015.1.01115.S & uid://A001/X2fb/X430 \\
		&  &  &  &  & & 5 & 25.7 & {\bf 0.22} & 3.1 & 2017.1.01301.S & uid://A001/X1273/X380 \\
		\hline
	\end{tabular}
	
	
\end{table*}

\subsection{Data reduction}
\label{sec:datareduc}

To obtain the calibrated $uv$-visibilities for imaging, we reduced the raw data using the default pipeline restoration scripts executed in the appropriate version of the Common Astronomy Software Applications (CASA) package \citep{mcmullin07}, i.e.\ the same version applied during the original calibration (CASA  {4.1} to {5.4.0-68}). Additional data quality checks and data manipulation were performed as described in the following section.

\subsubsection{Flux calibration accuracy}
\label{sec:fluxcalib}

We combine data from multiple objects and ALMA observation cycles. Because the addition of visibilities at incorrect flux scales could translate into erroneous spatial structure in the image plane,  we performed several quality assurance checks on the flux calibration.
As noted by \cite{stanley19}, several datasets required recalibration in a newer CASA version due to incorrect flux calibrator models present for Pallas and Ceres in CASA version 4.6 or lower. There are a total of nine datasets affected by this issue, as indicated in Table~\ref{tab:data}. Flux scales after recalibration were $60 - 80\%$ of the original values, with the mean flux level reduction of 15\% for these nine datasets. 

As a precaution, we reran the full pipeline calibration process on each of the 54 datasets, using the latest CASA version at the time of writing (version {5.6.1-8}), as described in the ALMA Science Pipeline User's Guide for CASA 5.6.1 (Section 6.4). Additionally, we manually enabled the query to the online ALMA flux calibrator catalog. 
The goal of this approach was to check for the existence of additional significant flux calibration changes introduced in the CASA development and debugging process, as well as to get the most up-to-date flux calibrator model values. For most of the datasets, the process finished successfully\footnote{Due to data format changes, CASA task definition changes, and backward incompatibility, this process failed for some datasets.} and yielded no flux level changes beyond 10\%, which is the usually reported calibration accuracy value. Because we found no further anomalies, we continued to use the calibrated visibilities provided by the default restoration scripts (except in the case of Pallas and Ceres calibrators as mentioned above).

\subsubsection{Data products per galaxy }
\label{sec:dataproduct}

We have  created several data products (measurement sets, maps and cubes)  in CASA for every observational cycle of each of the studied quasar (see Table~\ref{tab:data}), as described in \cite{venemans20}. We briefly summarize these here. 
We imaged the calibrated visibilities, using the CASA task {\sc tclean}, with the H\"{o}gbom deconvolver, and a channel spacing of 30~MHz (corresponding to 35~\ensuremath{\text{km\,s}^{-1}}\ on average). We weighted the data with the Briggs\footnote{We note that we recover the same aperture flux densities in both natural and Briggs weighted maps using the residual scaling method explained in a later section.} algorithm ({\sc robust = 0.5}) and cleaned (deconvolved) the maps down to $2\sigma$ level, within a cleaning mask radius of $2\arcsec$. We use these clean cubes to extract the \ensuremath{\text{\textsc{[C ii]}}}\ spectrum including the continuum emission. With the knowledge of the line position, we performed the continuum subtraction using the {\sc uvcontsub} task on the two spectral windows around the \ensuremath{\text{\textsc{[C ii]}}}\ line, excluding channels in total width of $2.5\times\mathrm{FWHM}$ around the \ensuremath{\text{\textsc{[C ii]}}}\ line (i.e.\ the total excluded bandwidth is on average 900\,\ensuremath{\text{km\,s}^{-1}}), while allowing for a continuum slope ({\sc fitorder = 1}). Using these datasets we imaged the continuum-free cube (same imaging parameters as above), as well as the \ensuremath{\text{\textsc{[C ii]}}}\ intensity map by averaging channels across $1.2\times\mathrm{FWHM}$ of the line, chosen to optimize the signal-to-noise ratio (S/N). We justify this integration width in Appendix~\ref{sec:app_optimalsn}.


\subsubsection{Stacking the data in the $uv$-plane}
\label{sec:uvstackmethod}

Stacking can be performed in the image plane by either averaging the  pixel/voxel values of maps or cubes, or by averaging extracted properties such as the spectra. Both methods are susceptible to several issues pertaining to the nature of interferometric data. I.e., the synthesized beam is not the same between datasets, and the faint stacked emission is usually uncleaned, which will yield ill-defined hybrid units in the stacked map (i.e., every pixel value will be a non-trivial combination of various cleaned and dirty beams). Another side-effect of image based stacking is that different spatial scales might be probed in different sources. In order to circumvent these issues,
we aim to add all visibilities in the $uv$-plane and thus obtain the best estimate of the mean emission of the population. To this effect we have adapted the method described in e.g.\ \cite{fujimoto19}, and extended it to be applicable to the whole spectrum. This method requires the data to be further reduced as follows.

For every individual cycle, starting from the continuum subtracted visibilities, the required channels were split out. Depending on the science goal, we either split out the equivalent of the \ensuremath{\text{\textsc{[C ii]}}}\ intensity map ($1.2\times\mathrm{FWHM}$), or, a smaller chunk corresponding to a specific velocity bin (i.e.\ a channel map, and/or the resolution of our spectral extraction). 
We used the function {\sc im.advisechansel} from the CASA toolkit to select the channels of interest in the kinematic local standard of rest frame (LSRK). Individual channels are selected based on the nearest neighbour algorithm. The data that we consider for $uv$-stacking were all observed at velocity resolutions of $\lesssim10\,\ensuremath{\text{km\,s}^{-1}}$, which is high enough to render any additional velocity-based data regridding or interpolation unnecessary. 
The data was further time-averaged inside 30\,s bins to reduce the data volume\footnote{The value is large enough to accomodate all datasets, but small enough not to introduce any time smearing effects.}. The coordinate reference system in the header was changed to the International Celestial Reference System (ICRS) as it was  sometimes incorrectly written as J2000. The visibilities were phase-shifted to the centroid of the \ensuremath{\text{\textsc{[C ii]}}}\ emission using the task {\sc fixvis}\footnote{The task uses small angle approximation, which is valid in our case as shifts are at most few arcseconds large, and less than an arcsecond in most of the cases.}. The header was modified further to artificially place the object at the $(0,0)$ coordinate\footnote{This was achieved by putting zeros into {\sc PHASE\_DIR, DELAY\_DIR,