Some banks include language in their deposit agreements that prohibits a convenience signer from closing an account. If there is not such a provision in the deposit agreement, the Uniform Commercial Code would permit any signer on an account to close it.
Even if the deposit contract does not permit a convenience signer to close an account out, he or she can bring the balance to zero, which may result in an eventual closing of the account if there is no subsequent activity. The surest way to get an answer to your question, of course, is to ask the bank.
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