Do conquer abilities trigger when you already held/conquered a battlefield, and what rules support this?

Ruling: Conquer abilities do not trigger when you already held/conquered that battlefield. Conquering is specifically a form of scoring, and you cannot score a battlefield again if you already scored it this turn.

Sequence:

• You can only score a battlefield once per turn
• Conquering requires scoring
• If you cannot score (because you already did), you cannot conquer
• If you cannot conquer, conquer abilities do not trigger

Nuances:

• You still gain control of the battlefield, but without scoring you cannot conquer
• Scoring includes both conquering and holding; inability to score means inability to do either