Unfortunately not since Valve is (understandably) keeping pretty quiet about how they implemented anti-cheat.

However, due to Wine/Proton always running as the current user, it is impossible for it to run anything in kernel space outside of the user-accessible part of the kernel API. Meaning it cannot install kernel modules, access memory of foreign processes or read anything your user does not have access to. It can't even get a full process list if the process does not want to be listed by users.

If you use Wine/Proton inside of Flatpak it cannot even read most of what your user has access to or any processes outside of the current Flatpak sandbox. So your Steam flatpak has no idea that you are running the Firefox flatpak on the same system with the same user.