Many questions of public interest do not have a single answer but come with a set of choices, each of which with its pros and cons.An “objective” information system can help explore the underlying argument space, and, if equipped with a conversational interface,it can create the experience of lively discussions resembling those from our daily lives. However, users will (subconsciously) extend the provided information by assumptions that adhere to their cognitive biases. In this regard, note that biases do not arise only from the underlying data or the employed algorithms, but also from the way the information is presented—especially in audio-only channels.Our paper brings attention to bias-related challenges of conversational interfaces for argument search systems. We identify research questions that address these challenges, and we propose ideas and methods to tackle them.