Signing out from PeopleSoft doesn't signout from other tabs when Public user setting is turned ON

We recently implemented Single Sign-on solution for our customer , as part of this implementation we had to turn the Public user setting ON at the web profile level.

Issue we are facing is:

When user opens multiple windows and sign out from any tab doesn't sign out other tabs instead when user goes to other tabs it opens public user session. Can we have solution that when Public user is enabled, signing out from one tab will automatically sign out from other tabs too instead of opening Public user session.

Note: Since the Public user session is used for Single Signon routing, we assigned a dummy role having no access to any components. And that gives an Authorization Error as expected, but this leads to bad user experience when users are sign out option while using multiple tabs.