I would start by asking if the user is sending the emails directly through Gmail’s web interface or via an email client. If the latter, there is a chance that the messages are not going through due to SMTP misconfiguration.
Perhaps the next step can be to ask the user to reply to a specific topic via email so that you can try to track their response down in your logs. If your community is busy, maybe you can create a private category specifically for this user to troubleshoot the email issue.