And by "fit" I mean will the cover go on top of it as it naturally would.
I use it with standard RPi cases above on the 3+, the lid will fit but it gets hot in the office so the fan runs all the time, so I run it with the lid off as it is our main DNS box and I want it close to the centre of the network. The machines in the basement have the lid on them and run ok.
Whilst the board has an extra layer on it, that layer doesn't extend past the USB sockets etc so the overall height is not larger.
I have a 4 running sans case but new case should arrive tomorrow but I don't expect it to be different as the changes in the case are for the new side connectors.