I believe the key issue here is the fact that a UGMA and a UTMA are both irrevocable accounts under the IRS code. So technically a customer should not transfer funds from a UGMA to an UTMA and vice versa.
Based on your question above, the account would need to be titled UTMA if your state does not recognize UGMA's. The customer transferring the funds from the UGMA to a UTMA would be responsible for the tax implications, not your bank.
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The opinion stated here is what it is, My Opinion.