If I conquered a battlefield on turn 2, scored it and moved my unit back on turn 3, and my opponent didn't conquer it on their turn 3, do I conquer and score it again when I move a unit there on my turn 4?

Ruling: Yes, you would conquer and score the battlefield again on turn 4.

Nuances:

• When turn 4 starts, the battlefield is uncontrolled (not yours), even though you previously controlled it
• You can only score each battlefield once per turn
• You can only conquer if you score