I’ll check this on my monitor - I run about 5 virtual desktops on that monitor.
Are you running different apps on virtual desktop 2 than virtual desktop 1? And different apps on 3 than you are on 2?
If your system preferences desktop space settings are correct and your apps are set to “none” (when you right-click on app in dock and select options > assign to) then your apps will open ONLY on the virtual desktop they were originally opened on — resulting in different apps running in different virtual desktops.
In that scenario, once you have multiple apps running in any of your virtual desktops, and stage manager is turned on and set to show apps in sidebar, then the auto-hide should work.
I’ll double-check when I’m back at my desk.