This paper provides a foundational analysis of the regulatory obligations imposed by the European Union Artificial Intelligence Act (EU AIA) on conversational agents (CAs) that integrate large language models (LLMs) via APIs. Addressing a significant gap in existing research, we systematically map the relevant compliance requirements and clarify the categorization and risk classification of such systems under the EU AIA. We present a preliminary compliance framework as an initial artifact, serving as a basis for future design science research (DSR) aimed at developing actionable technical guidelines and empirically validated design principles. While this study does not offer a fully developed or evaluated artifact, it establishes the problem space and regulatory landscape, setting the stage for subsequent research phases that will translate legal mandates into concrete design features for EU AIA-compliant conversational agents.