As LLM-based Conversational Avatars increasingly act as collaborators in hybrid indoor navigation, understanding how their personality traits influence human-avatar proxemic behavior is becoming crucial. Prior work has largely examined personality effects in static or one-sided interactions such as sitting, standing, or approaching. However, there is a gap in research on how avatar personality and motion-related factors (e.g., walking speed) shape proxemics when both the human and avatar are in motion. To address this, we developed an AR indoor navigation system featuring a Conversational Virtual Avatar (CVA) with three distinct personalities: Dominant, Warm, and Conscientious. The CVA guides users to destinations within the environment. In a between-subjects study (N=27), we found statistically significant effects of avatar personality and walking speed on proxemic behavior. Our work contributes to a broader understanding of the role of personality and walking speed of a CVA on human-avatar proxemic behaviour during navigation.