Does Blighted Battleaxe require the equipped unit specifically to conquer, or does any conquest by the controller prevent its end-of-turn effect?

Ruling: The equipped unit specifically must conquer a battlefield to prevent the effect. If that particular unit did not conquer during your turn, Blighted Battleaxe unattaches and deals 4 damage to the equipped unit.

Nuances:

• When gear is attached, its Effect Text becomes part of the equipped unit's rules text. The pronoun "I" or "me" in the gear's effect refers to the equipped unit, not the gear itself.
• "This" in the effect refers to the gear (Blighted Battleaxe).
• Conquests by other units you control do not satisfy the condition for the equipped unit.