Pedagogical Conversational Agents (PCAs) conquer academia as learning facilitators. Due to user heterogeneity and need for more inclusion in education, inclusive PCA design becomes relevant, but still remains understudied. Our contribution thus investigates the effects of inclusive PCA design on competence, trust, and diversity awareness in a between-subjects experiment with two contrastingly designed prototypes (inclusive and non-inclusive PCA) tested among 106 German university students. As expected by social desirability, the results show that 81.5% of the probands consider an inclusive design important. However, at the same time, the inclusive chatbot is highly significantly rated as less competent. In contrast, we did not measure a significant effect regarding trust, but a highly significant, strongly positive effect on diversity awareness. We interpret these results with the help of the qualitative information provided by the respondents and discuss arising implications for inclusive HCI design.