If a player already scored a point for holding a battlefield at the start of their turn, then leaves it empty (making it neutral) and moves a unit back onto it in the same turn to conquer it, do they score another point?

Ruling: No, you cannot score another point. Each battlefield can only be scored once per turn, regardless of whether you score from holding it or conquering it.

Nuances:

• Even if a battlefield becomes neutral mid-turn after you already scored from holding it, moving back onto it does not grant an additional point in the same turn.