 related with some differences between our analysis, as some dependence with the fiducial values. In any case, we have obtained a more conservative result.
\begin{figure}
    \centering
    \includegraphics[width=0.95\columnwidth]{M3_1p515.pdf}
    \caption{Case $Q \propto \rho_c$ ($\xi \equiv \xi_c$). The ellipses
        represent the 68 per cent uncertainty around the fiducial
        \gls{lcdm}\ model. The thin and thick lines correspond to results
        considering the survey areas \SIlist{4000;8500}{\square\deg},
        respectively. The red contours are for \gls{lrg}, green for \gls{elg},
        blue for \gls{qso}, and black for the Multi-tracer analysis,
        all with \gls{rsd}.}
    \label{fig:M3joint}
\end{figure}

\begin{table}
    \setlength\tabcolsep{6pt}
    \renewcommand*{\arraystretch}{1.2}
    \caption{Marginalized uncertainties for the three surveys, without
        \gls{rsd}, for the case where the interacting coupling term is proportional to the sum of the dark sector energy densities, $Q \propto \rho_c + \rho_d$ ($\xi_c = \xi_d \equiv
        \xi$).}
    \label{tab:margerrors_model4_noRSD}
    \centering
    \sisetup{
        round-mode=places,
        round-precision=3,
        table-format=1.3
    }
    \begin{tabular}{@{}lcSSSS@{}}
        \hline
        &   {Uncertainty}   &   {\gls{lrg}} & {\gls{elg}} & {\gls{qso}} & {Multi-tracer} \\
        \hline
        \multirow{3}{*}{\begin{minipage}[t]{0.45 in} \gls{jpas} \\ {\footnotesize (\SI{4000}{\square\deg})} \end{minipage}} %
        &   $\sigma_{\Omega_d}$  &   0.707856 &   0.670141 &   0.353694    &   0.203799    \\  
        &   $\sigma_{w}$    &   1.14623 &   1.10284 &   0.794619    &   0.395584    \\  
        &   $\sigma_{\xi}$  &   0.58476 &   0.528425 &   0.216951    &   0.132972    \\
        \hline
        \multirow{3}{*}{\begin{minipage}[t]{0.45 in} \gls{jpas} \\ {\footnotesize (\SI{8500}{\square\deg})} \end{minipage}} %
        &    $\sigma_{\Omega_d}$ &   0.697195   &   0.6227  & 0.259794    &   0.142753 \\
        &   $\sigma_{w}$    &   1.11693 &   1.02155 &   0.583123    &   0.27704    \\
        &   $\sigma_{\xi}$  &   0.572847     &   0.490286    &   0.15925    &   0.0931193   \\
        \hline
        \multirow{3}{*}{DESI} &   $\sigma_{\Omega_d}$ &   0.67887 & 0.406324    & 0.578443    & 0.283681    \\
        &   $\sigma_{w}$    &   1.1041 & 0.70097    & 1.0689 & 0.489726    \\
        &   $\sigma_{\xi}$  &   0.545598 & 0.304971    & 0.411653    & 0.212106    \\
        \hline
        \multirow{3}{*}{Euclid} &   $\sigma_{\Omega_d}$  &       &   0.185087    &     &     \\
        &   $\sigma_{w}$    &       &   0.336319    &     &      \\
        &   $\sigma_{\xi}$  &       &   0.131775    &     &    \\
        \hline
    \end{tabular}
\end{table}
\begin{table}
    \setlength\tabcolsep{6pt}
    \renewcommand*{\arraystretch}{1.2}
    \caption{Marginalized uncertainties for the three surveys, with
        \gls{rsd}, for the case where the interacting coupling term is proportional to the sum of the dark sector energy densities, $Q \propto \rho_c + \rho_d$ ($\xi_c = \xi_d \equiv
        \xi$).}
    \label{tab:margerrors_model4_RSD}
    \centering
    \sisetup{
        round-mode=places,
        round-precision=3,
        table-format=1.3
    }
    \begin{tabular}{@{}lcSSSS@{}}
        \hline
        &   {Uncertainty}   &   {\gls{lrg}} & {\gls{elg}} & {\gls{qso}} & {Multi-tracer} \\
        \hline
        \multirow{3}{*}{\begin{minipage}[t]{0.45 in} \gls{jpas} \\ {\footnotesize (\SI{4000}{\square\deg})} \end{minipage}} %
        &   $\sigma_{\Omega_d}$  &   0.050504   &   0.0301982   &   0.0575123   &   0.0188978  \\  
        &   $\sigma_{w}$    &   0.17809    &   0.0944343   &   0.141167    &   0.0410643  \\  
        &   $\sigma_{\xi}$  &   0.110027   &   0.0349267 &   0.0405026   &   0.0187075  \\
        \hline
        \multirow{3}{*}{\begin{minipage}[t]{0.45 in} \gls{jpas} \\ {\footnotesize (\SI{8500}{\square\deg})} \end{minipage}} %
        &    $\sigma_{\Omega_d}$ &   0.0346917   &   0.0207315   &     0.0395224   &   0.0129814   \\
        &   $\sigma_{w}$    &   0.122716   &   0.0650148   &   0.0970109   &   0.028301   \\
        &   $\sigma_{\xi}$  &   0.07584   &   0.0240049   &   0.0278224   &   0.0128428   \\
        \hline
        \multirow{3}{*}{DESI} &   $\sigma_{\Omega_d}$ &   0.0235797   & 0.0120747   & 0.0337923   & 0.0109078   \\
        &   $\sigma_{w}$    &   0.0934715   & 0.0476072   & 0.126902    & 0.0383423   \\
        &   $\sigma_{\xi}$  &   0.0539899   & 0.012611   & 0.0361464   & 0.0106951   \\
        \hline
        \multirow{3}{*}{Euclid} &   $\sigma_{\Omega_d}$  &       &   0.00761294    &     &     \\
        &   $\sigma_{w}$    &       &   0.024792     &     &      \\
        &   $\sigma_{\xi}$  &       &   0.00586946     &     &    \\
        \hline
    \end{tabular}
\end{table}
For the last case considered, $Q \propto \rho_d + \rho_c$, we give the results for the marginalized constraints in Tables~\ref{tab:margerrors_model4_noRSD} (without \gls{rsd}) and \ref{tab:margerrors_model4_RSD} (with \gls{rsd}).
As one could expect, in this scenario we see characteristics combined from both
previous cases. The measurements of $H(z)$ and $D_A(z)$ give significant information,
especially at high redshifts, as in $Q \propto \rho_c$. 
The constraints are also very sensitive to \gls{rsd} as in $Q \propto \rho_d$.

