Can a stunned unit conquer an open battlefield

Yes. A stunned unit still conquers an open (empty and uncontrolled) battlefield — Stun only stops a unit from contributing Might to combat damage (Rule 423.1.b); it doesn't affect establishing Control after a Non-Combat Showdown (FAQ 3409).

The contrast: if enemy units are present and survive combat, the stunned attacker is recalled during Combat Cleanup (3d) instead of conquering. Against an open battlefield there's no combat at all — just a Non-Combat Showdown where your unit alone establishes Control and Conquers.