Trends between the fraction of carbon mass in the cold gas phase (T < 5 × 10⁴ K) in the diffuse halo (f_C) and the distance (r) between the galaxies (left), the stellar mass ratio (μ_⋆, middle), and the carbon mass ratio (μ_C, right). In particular, on the left, we show f^avg (i.e., the average among the three projections) as a function of both the distance averaged on the three projections (r_m, violet squares) and the physical distance between the galaxies (R, violet stars). In the middle and right panels, different colors refer to f_C estimated by projecting the carbon surface density along the lines of sight parallel to the z (blue), y (pink), and x (red) axis. In each panel, white dots are f^avg, the black line is the linear fit considering f^avg versus r_m, μ_⋆, μ_C, respectively. We also specify the Spearman coefficient (ρ) of each relation. H2 is not included in these plots as this system is characterized by an out-of-scale value of f_C = 0.8 and H6 is highlighted in gray. The horizontal gray dot-dashed line shows the value of f_C = 0.5.