Understanding the physical connection between cluster galaxies and massive haloes is key to mitigating systematic uncertainties in next-generation cluster cosmology. We develop a novel method to infer the level of conformity between the stellar mass of the brightest central galaxies~(BCGs) $M_*^{BCG}$ and the satellite richness $lambda$, defined as their correlation coefficient $rho_{cc}$ at fixed halo mass, using the abundance and weak lensing of SDSS clusters as functions of $M_*^{BCG}$ and $lambda$. We detect a halo mass-dependent conformity as $rho_{cc}{=}0.60{+}0.08ln(M_h/3{times}10^{14}M_{odot}/h)$. The strong conformity successfully resolves the halo mass equality conundrum discovered in Zu et al. 2021 --- when split by $M_*^{BCG}$ at fixed $lambda$, the low and high-$M_*^{BCG}$ clusters have the same average halo mass despite having a $0.34$ dex discrepancy in average $M_*^{BCG}$. On top of the best--fitting conformity model, we develop a cluster assembly bias~(AB) prescription calibrated against the CosmicGrowth simulation, and build a conformity+AB model for the cluster weak lensing measurements. Our model predicts that with a ${sim}20%$ lower halo concentration $c$, the low-$M_*^{BCG}$ clusters are ${sim}10%$ more biased than the high-$M_*^{BCG}$ systems, in excellent agreement with the observations. We also show that the observed conformity and assembly bias are unlikely due to projection effects. Finally, we build a toy model to argue that while the early-time BCG-halo co-evolution drives the $M_*^{BCG}$-$c$ correlation, the late-time dry merger-induced BCG growth naturally produces the $M_*^{BCG}$-$lambda$ conformity despite the well-known anti-correlation between $lambda$ and $c$. Our method paves the path towards simultaneously constraining cosmology and cluster formation with future cluster surveys.
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