I played yesterday, it was amazing! It was a bit of a hustle to launch it but I did it!

Here is what helped me: first, there is some kind of mess about nvidia module, i have no clue what this command does, but it helped! sudo nvidia-modprobe -u -c=0

Then, instead of seeing some error on unreadable window the game went launching but not showing anything. So, to make everything work I shoved battle.net app into steam.

Then I have downloaded this tar.gz and unpacked it in /home/<username>/.steam/root/compatibilitytools.d/ compatibilitytools.d folder wasn't there, had to create it

After that I had Proton-6.16-GE-1 as a compatibility environment in my steam, and I forced it to be used with battle.net app and it worked! The game worked perfectly!