For me, I think it might in part depend on what I thought of the ex-girlfriend. As you know, my present partner has a terrible ex. I would likely insist on some sort of symbolic and/or practical cleansing if they offered me something that had belonged to the ex or been used extensively by the ex even if there hadn't been ownership.

On the other hand, I have the deepest respect and affection for my (now ex) Figment's late wife. If Figment had etc., I would have been deeply honored (and probably run away screaming because we wouldn't have made good spouses).