Using multiple users and external monitors confuses MacOS

I have a "work" account as well as a "private" account on my m1 MacBook. Disconnecting from an external monitor works on the "active" account. However, when switching to the second account, it seems it still "thinks" it has the external monitor connected and all windows are still arranged accordingly. This means windows are "lost" and also the taskbar is sometimes not appearing. Switching the the other account and back again fixes this. I run Ventura 13.3 but noticed it as well on earlier versions.