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Yes, it will show a balance unless/until they sell it to a collection agency or law firm. Then, it should update to a $0 balance, but of course, then you will have an additional derogatory from the collection agency and they will show the balance and very likely additional fees.
Yes, that is what I would do. It will save you from the possibility of the future derog from a collection agency, possible lawsuit, and it will help your UTIL when the account is paid. They will also most likely stop updating the monthly CO, which hurts your score each time they do it.
Good luck!
Let us know how it goes. Good luck!!