Comparison with simulations. Left: Mass vs. redshift for literature proto-clusters and TNG300 simulations. The gray and blue shaded areas show the simulated halo mass evolution of proto-clusters in the Chiang et al. (2013) and TNG300 simulations (Montenegro-Taborda et al. 2023), repectively. The red shaded area marks the stellar mass of BCGs in TNG300 (Montenegro-Taborda et al. 2023). The halo mass of peak A and the stellar mass of Galaxy A are consistent with the predictions from the models, suggesting a massive descendant with a halo mass of more than 10¹⁴ M_⊙ at z = 0. Right: sSFR vs. redshift for BCGs. The sSFRs of BCGs in TNG300 simulations (Montenegro-Taborda et al. 2023) are shown as blue and red curves, overlaying with the observed sSFRs of massive quiescent members in z > 2 proto-clusters. The blue shaded regions indicate the 16th to 84th percentile range of the r < 50 kpc sSFR measurements in TNG300.