I have 2 artifacts sets stored. One is equipped. The other, I cannot equip. I did a item search and the items don't exist.

How is this possible? Did I delete/market/vendor them? They don't get removed from sets?

In any case, I am unable to remove this set or clear out the stones, nothing. Feels like, yet another, broken or terribly thought out system.

Do you have the artefact set equipped on another character? You have to store it in the artefact inventory again if you want to use it on other characters.