The filter bubble is a notorious issue in Recommender Systems (RSs), characterized by users being confined to a limited corpus of information or content that strengthens and amplifies their pre-established preferences and beliefs. Most existing methods primarily aim to analyze filter bubbles in the relatively static recommendation environment. Nevertheless, the filter bubble phenomenon continues to exacerbate as users interact with the system over time. To address these issues, we propose a novel paradigm, Hypergraph-Aware Multi-Grained Preference Learning to Burst Filter Bubbles in Conversational Recommendation System (HyperCRS), aiming to burst filter bubbles by learning multi-grained user preferences during the dynamic user-system interactions via natural language conversations. HyperCRS develops Multi-Grained Hypergraph (user-, item-, and attribute-grained) to explore diverse relations and capture high-order connectivity. It employs Hypergraph-Empowered Policy Learning, which includes Multi-Grained Preference Modeling to model user preferences and Preference-based Decision Making to disrupt filter bubbles during user interactions. Extensive results on four publicly CRS-based datasets show that HyperCRS achieves new state-of-the-art performance, and the superior of bursting filter bubbles in the CRS.