Does Nintendo Labo work on my switch?

I wanted to buy the Nintendo Labo Toy Con 1 set for my Switch, so the very first one with the piano etc. However, my Joycons are in a case that has a small increase on the back to improve the feel. Do the Joycons fit anyway? And if not, can I remove a little cardboard and then it fits?

Can't you take off and put on your covers? Mine can be easily removed.

(If you don't get a clear answer, I would still invest the 4.99 euro and try to adapt the cardboard at the point with a gentle crease or even cut it. I wouldn't risk it with a larger labo set.)

Thanks for the asterisk x)