I did, in fact, do this after I received the bill for garbage from June through July. The person I talked to said he'd clear that out, and that I owed nothing.
I just received a bill for that amount again, plus late fees, as well as charges for August and September. Apparently whatever the first guy did went nowhere.
I believe this is the same finance department that seems to have decided that (1) paying my business tax, followed by (2) changing my address several months later means that (3) I'm operating a business without a license.
I just so love dealing with morons.