  We use deep high resolution 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 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.
