You are in powersave mode if your connections are terminated after 30 secs, which is the default timeout if there is no data being transferred.
You are mixing concepts here. Background connections have nothing to do with connection disconnection. Your email application is keeping connections up because there is data being transferred so powersave idle timer is not triggered.
It Is possible to configure the system to disconnect inactive connections or if application quits. No one has just not yet created an app that does this kind of system configuration. Note that the default system ui could get confused by this as it is made to support only the always online use case.