Can one bluetooth device (mouse) automatically connect to alternating devices (2 windows laptops) depending on whether they are up?

I want to use my bluetooth mouse (Rapoo) with any of two laptops (running windows 8 and 10, resp.), depending on which laptop is powered up (I have only one up at any time). What I would like to have is this (any sequence of the first 3 steps is good :-) turn on the laptop turn on the mouse press bluetooth pairing button on mouse running laptop connects Right now, the only procedure which works halfway predictably is to turn on both devices go to laptop's bluetooth menu disconnect mouse (yes, it's shown as connected, although it was connected to the other laptop before, and it doesn't work wit

Question by virtualnobi on Super User, licensed CC BY-SA. We show a short excerpt and link to the original thread as credit.