Do quest items despawn when the person quits the game?

Solution 1:

Yes you are exactly right. The item you are talking about is a Tall Tale item for "The Revenge of the Morningstar" and this skull along with another item is used to spawn the boss. Since it is not an item you can actually sell it de-spawned since they left but the chest stays because you can continue to use it like the regular collectors chests. You can also sell the chest.