The SG CachePress plugin is created to handle the SuperCacher service which combines an NGINX reverse proxy and Memcached. I’ve checked your ticket history and the issues you’re experiencing are connected with the Google PageSpeed module.
The SG CachePress plugin does not clear that cache and does not interfere with Google PageSpeed in any way. The PageSpeed module is listed in the SuperCacher section to provide you with an easy interface to stop and start it. This is why our support team has proposed installing the entire SuperCacher module for your server. In addition to that, you will notice that once you enable PageSpeed the Dynamic and Static caches for that domain get disabled.
So to sum up the SG CachePress plugin does not handle Google PageSpeed. If you use the plugin and the dynamic caching I can assure you that no logged in users will be served with cached content.