

It works fine.

The framerate is all over the place when graphic settings are set to anything higher than Medium. The game still looks great with that setting, so I don't consider this a hugh problem.
I played with whatever the default Steam Deck configuration was as of March 2024 (default Proton and controller config), graphics set to Medium, and VSync off. Everything seemed to look and run well!
Haven't played it very far yet, but so far no issues whatsover.
Doesn't work at all on steam deck
Shenmue 3 is not a particularly optimized game, so the framerate issues aren't a surprise on Very High and High settings. Medium is the way
The game is playable, but with massive framerate shifts on Very High and High settings. Can get close to 60 fps in spots but will also drop as low as 20 fps, with an average around 30 fps. If you change the settings to Medium and use the SteamOs Beta to drop the refresh rate to 40 hrz and limit the framerate to 40 fps, it provides a pretty smooth experience early in the game.
As mentioned before, large framerate variations if playing on Very High or High Settings. Good if playing on Medium at 40 hrz and 40 fps.
Just like a native game.
When you first start the game, it will stutter until the shader cache is complete, shouldn't be more than a few minutes. After than, it's perfect.
Works perfectly fine.
It runs out of the box though users might want to switch to Proton-5.21-GE-1 if using a gamepad.
Had to use Proton-5.21-GE-1 in order for my Gamepad to work (DS4 via Steam Input)
Proton-5.21-GE-1
Framerate sometimes tanks after opening menus, but can be easily fixed by saving and reloading.
Works flawless directly with Steam via Proton

Runs perfectly out of the box.
EGS is no longer necessary.

Other than the screensaver from KDE not being disabled automatically, I did not perceive any issues whatsoever compared to Windows. Given I had Windows on my machine only for this game, I'm confident I can now uninstall it. This is my first time trying Proton/Lutris, and I'm very impressed on what's been achieved.

Installed Epic Game Store with Lutris
Install Epic Game Store (As it isn't on Steam Yet) via Lutris and it simple works.