I was also unable to reproduce this issue. I'm testing with Drupal 9.5, the latest dev version of automated logout, and a stable version of OpenID Connect configured to use the GitHub client.
I've tested using Firefox and Chrome. Both just closing the tabs, fully closing the browser, or waiting for the timeout to occur. I simply can not reproduce it.
Could you please provide some more information? What OpenID connect client are you using? How did you configure the Automated logout module?
Postponing the issue for now.