For your consideration:
You might consider leaving the question attached to the profile if it provides information that will be of use to others who are perusing your profile.
For example, in the Great Migration work I do here, I frequently create a question along the lines of: "Who were the parents of immigrant John Smith?" and it's linked to his profile. Even if the question is answered, I retain the link between the profile and the question so that others can see the correspondence that relates to the question. This is particularly useful when there are disputed origins or spouses.