This study evaluates the effectiveness of conversational agents (CAs) in eliciting sensitive information, focusing on how their design—specifically conversational relevance—influences user disclosure. This research also examines the role of individual differences, such as social desirability and sensitivity, in moderating this relationship. Utilizing a vignette-based factorial experiment, we evaluate the proposed conceptual model. The findings suggest that while conversational relevance generally enhances information disclosure, its positive effect is negated in users prone to socially desirable responding or exhibiting high sensitivity. This insight is central for the optimization of CAs, highlighting the need to consider individual psychological traits in their design and deployment, especially in contexts requiring the elicitation of sensitive information.