New Battle.net update Version 14542 breaks Linux compatibility with Wine/Proton via Lutris or Steam and will only display a gray screen.
Workaround is to delete the update folder Battle.2.net.14542 under $wine_root$/drive_c/Program Files (x86)/Battle.net/ and then start the client again.
Update workaround: Set WINE_SIMULATE_WRITECOPY=1 environment variable in Lutris
If you are using Lutris, they indicated that adding "WINE_SIMULATE_WRITECOPY=1" environment variable should work (cf. https://fosstodon.org/@lutris/111503016713557868).
Thanks for the information and the link. Updated the opener.