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Conversational AI services are rapidly expanding, yet most systems still generate responses based on intent classification, sentiment detection, and static persona prompts. In Korean—a representative high-context culture—indirect expressions, honorifics, hesitation, and relational distance critically shape user experience but remain difficult to capture through surface-level analysis. This study proposes a nunchi-based conversational UX framework for Korean-language AI companion agents. Employing literature review, comparative service analysis, and case-based design, the framework comprises five layers—Input, Nunchi Interpretation, Margin Control, Response Policy, and Agent Output—and was applied to the XenLook On agent dialogue service. The findings indicate that: (1) Korean conversational experience can be structured as a cultural design problem rather than solely an NLP problem; (2) nunchi operates as a hierarchical UX model regulating inference, clarification, and neutral responses through a belief margin mechanism; and (3) the framework reduces over-familiarity and over-interpretation while enhancing interaction continuity. The proposed framework offers a practical reference model for designing culturally sensitive AI companion agents.