No, you cannot hold the customer liable. You are contractually obligated as a VISA issuer to provide all of the protections offered by VISA.
18.104.22.168 Zero Liability – US Region (Updated)
In the US Region, an Issuer must limit the Cardholder’s liability to zero upon receipt of notification from its Cardholder of an unauthorized Visa Consumer Card or Visa Business Card Transaction.
The Issuer may increase the amount of the Cardholder’s liability for unauthorized Visa Transactions only if the Issuer reasonably determines, based on substantial evidence, that the Cardholder was grossly negligent in the handling of the Account or the Card.
A Visa Business Card Issuer may require the notification of unauthorized Visa Transactions to be received within 60 calendar days of the mailing date of the first statement showing unauthorized Visa Transactions.
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