I doubt there was a duel after Eärnur and his men were captured since he was supposedly a beast and no man in Gondor could even match him. His prowess was such that none in Gondor could stand against him in those weapon-sports in which he delighted, seeming rather a champion than a captain or king. (Gondor and the 2023 Ncaa Division I Mens Basketball The Road To Houston March Madness 1st 2nd Rounds Albany Long Sleeves T Shirt and I will buy this Heirs of Anárion) So there was no Witch-king and Eärnur fighting it out once and for all imo. I think the Witch-king had the King and his company killed. If they were captured alive for later torture it would not have been easily done. It is not known what happened in the end since, “there were no witnesses of his death”, but as I said I doubt the Witch-king was even considering fighting him to single combat due to the powerful nature of the man. In Gondor they thought “that the faithless enemy had trapped the king, and that he had died in torment in Minas Morgul” but this is just a guess on their part. I think he probably died fighting against being captured.