I cleared the cave first. Then learned of the quest & spoke to the king.
When I tried to turn it in, it wouldn't let me. When I told him I couldn't complete it and the quest failed - it THEN gave me the option to turn the horn in. And forced me to retake the quest.... which sent another troop of Iniau at me. Didn't mind that battle a second time, it was fun and had good loot.
Does the cave ever reset?