We use deep high resolution textit{HST/ACS} imaging of two fields in the core of the Coma cluster to investigate the occurrence of nuclear star clusters (NSCs) in quiescent dwarf galaxies as faint as $M_{I} = -10$ mag. We employ a hierarchical Bayesian logistic regression framework to model the faint end of the nucleation fraction ($f_{n}$) as a function of both galaxy luminosity and environment. We find that $f_n$ is remarkably high in Coma: at $M_{I} approx -13$ mag half of the cluster dwarfs still host prominent NSCs. Comparison with dwarf systems in nearby clusters and groups shows that, within the uncertainties, the rate at which the probability of nucleation varies with galaxy luminosity is nearly universal. On the other hand, the fraction of nucleated galaxies at fixed luminosity does exhibit an environmental dependence. More massive environments feature higher nucleation fractions and fainter values of the half-nucleation luminosity, which roughly scales with host halo virial mass as $L_{I,f_{n50}} propto mathcal{M}_{200}^{-0.2}$. Our results reinforce the role of galaxy luminosity/mass as a major driver of the efficiency of NSC formation and also indicate a clear secondary dependence on the environment, hence paving the way to more refined theoretical models.
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