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Since the proposal of the Weak Cosmic Censorship Conjecture, numerous studieshave emerged. In this paper, we investigate by injecting a test particle and ascalar field into a Kerr-like black hole immersed in a dark matter halo. Weconsider both first-order and higher-order cases and find that Kerr-like blackholes immersed in a dark matter halo do not violate the Weak Cosmic CensorshipConjecture. However, when a test particle is dropped in, there exists apossibility in the near-extremal condition where the event horizon of the blackhole can be destroyed. Overall, our analysis suggests that the presence of adark matter halo hinders the super-spinning of black holes, implying that colddark matter seems to play a protective role. Without considering cold darkmatter, our conclusions are consistent with previous research.

Exploring the Impact of Coupled Behavior on the Weak Cosmic Censorship Conjecture in Cold Dark Matter-Black Hole Systems