Currently, the Wine and Proton compatibility layers for Linux don't work with Easy Anti-Cheat and we have something of an update on the status for you.
Easy Anti-Cheat is one of the most widely used systems to reduce cheating in games, it's available in some form for actual Linux builds of games but it's something of a sore spot for Wine and Proton. Some time ago, it was confirmed that Valve and the Easy Anti-Cheat team were planning to work together to get the situation sorted, Epic Games later confirmed Easy Anti-Cheat was still supported on Linux for native builds too after it appeared that was stopping. Since then, we've not really heard anything officially on it.
However, over on Reddit, user Guy1524 who happens to work for CodeWeavers (who work on Wine / Proton) gave a personal update on their own clearly unofficial (EAC themselves are not involved) progress to get Easy Anti-Cheat working.
At this point, EAC will load, correctly process an IOCTL in which the loader sends an encrypted "internal dll", which it expects the driver to inject into the game process. EAC will then correctly map this dll when the process loads, and try to initialize it. This is the point where it is failing right now (the internal dll will yield an error), but we seem to be really close to the end of the initialization function, and once blitzcrank finishes devirtualizing it, we should be able to quickly get the function to succeed. In theory, if all goes to plan, the game should then launch, and be able to load the EAC library through the hooks setup by the internal DLL / kernel.
Wine (and so Proton) has always been a game of cat and mouse for Windows games on Linux, a constant game of catch-up as developers do new and different things and break compatibility. Anti-cheat tools add another layer of complexity that can cause more problems. The issue here, is that even if they manage to get it hooked up and working, EAC could end up doing something to block it if they don't like how it's operating. This is actually something that Epic Games Founder and CEO, Tim Sweeney, mentioned on Twitter when asked about it:
We'd be fully supportive of these efforts if confident they wouldn't lead to the worst-case scenario, which is a significant increase in cheating that we have no ability to detect.
If they manage to get Easy Anti-Cheat properly working with Wine and Proton, it would open up yet another big world of gaming on Linux. I think we can all agree that would be a great thing until the day our market share rises enough that more game developers support Linux directly.