In the Stack Exchange user reputation leagues page, if I select any site in the site switcher it will be set as Stack Overflow or some random site in the drop down list's selected site name.
For the Stack Overflow, Super User, Server Fault and Meta Stack Exchange, the site switcher is set to the selected site name, but the remaining do not retain the selected site name. For English Language and Usage, it changes to Physical Fitness for some reason. All the others I've tried switch to Stack Overflow.
It can be replicated in Chrome and Firefox.
GIF for the same: