Yes it would still apply. Also, you did not mention if it was a joint account in which case regardless of whose card was used, either account owner can dispute a transaction under Reg E. Although your debit card processor will require a written letter from the cardholder in order for you to file a chargeback, you are still obligated under Reg E to investigate even without the cardholder letter.
In the event you are unable to obtain a cardholder letter to file a chargeback, you can always file a retrieval request which requires the merchant to provide the same information as a chargeback. The only difference is that the bank does not receive provisional credit the way it does with a chargeback. At least you will receive documentation to make a determination on the dispute. Also if the merchant fails to respond within 30 days of your request, you can file a chargeback without a cardholder letter.
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