Feedback Genius does not function as an auto-reply system, or mark auto-reply responses as "responded."  Feedback Genius only handles the outgoing emails sent by redirecting emails to Seller Central, and the emails technically sending on your behalf.  Everything that happens on Seller Central we do not have any control over.
 
The only work around is to add a customer onto your email blacklist if they've started communication with you.  This prevents automated messages from Feedback Genius to be sent to a particular customer.
 
Even without using a service like Feedback Genius, organic emails through Amazon may also generate auto-replies.  We feel that there is value in addressing this with Seller Support to see of a more optimized way of handling these cases.
 
Here is a useful Amazon article on the subject, "Contact Response Time Metrics."

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