In a 2v2 game, if my teammate controls a battlefield at my beginning phase (disqualifying me from scoring it), but then their units leave and I move units there and win the showdown, do my conquer abilities trigger?

Ruling: No, conquer abilities do not trigger because conquering is a subset of scoring, and you are disqualified from scoring that battlefield this turn.

Nuances:

• Rule 376.4.b.2.c (about conquer effects triggering even when the point gain is negated/replaced) mainly applies to situations like drawing 1 card instead of gaining the final victory point, where you are still deemed to have scored/conquered the battlefield
• The teammate control disqualification prevents scoring entirely, so conquering does not occur