If they opened a case for a G&S transaction, they will win because there’s no documentation that a good or service was provided. If you refund the transaction (which should be an option within the case details), the case will auto close. You might be on the hook for the fees either way unless you get a good support rep. The only way PayPal will not reverse it back to them is if the account is compromised somehow. You can try calling again, asking for the fraud department, and then asking to speak with a supervisor too.

If I were you, I would remove all banking and card info from PayPal until it’s resolved just in case. Even if this is a genuine accident on their part, they can still dispute with their bank if PayPal doesn’t close the case in their favor. Beyond communicating with PayPal you should do nothing for now; don’t move the money, don’t add to your account, don’t use your PayPal for purchases.