The self-signed certificates on a Hostgator dedicated server were nearing expiry, and notification emails like 'The SSL certificate for xxx on yyy.com will expire in less than 30 days' were starting to appear.
After going through domain verification on chat, found that they don't do SSL install support on live support. You need to install a new certificate as soon as possible. You can do this with WHM's Manage Service SSL Certificates interface at
https://yyy.com:2087/scripts2/manageservicecrts.