You can set up a script using the lastActivity session prop (I'd probably use a property bound to the current date, updating once per minute, with a change script like in this post) and then call the system.security.switchUser() function to switch to some "guest" user, since it seems like the logout function takes you to the login page again.
You could probably also have a script that automatically logs the user in to the "guest" user on session start, and hijack the logout button to switch user to your "guest" instead. never mind, can't do that in Perspective
Maybe someone else has a less complicated solution solution that actually works though.