Halo mass assembly history: in the upper panel we show the redshift evolution of median values of the halo mass, M_vir , according to the selection procedure outlined in Fig. B.1 for our five selected sub-samples: low-SFR (solid blue line with white dots indicating the redshift values of each snapshot), passive (solid light yellow line), red (short-dashed red line), low-Z_cold (solid dark magenta line), and high- Z_cold (dotted-dashed green line). The shaded regions represent the range spanning between the 32^nd and the 68^th percentile around the median. Their corresponding mass growth history relative to the reference redshift of our study, z_ref = 0.56, as z₀ . In this panel, X_z denotes the values of M_vir at a specific snapshot/redshift compared to the values at z₀ (${\text{X}}_{{\text{z}}_0}$). The horizontal dashed black line indicates the threshold of 50% of the total halo mass at z_ref.