I'm attempting to play Tabletop Simulator in VR, on Linux. I've installed the game via Steam.

If I install the native version, the game correctly launches in 'flat' mode, and the tutorial can be played.
If I launch the game while SteamVR is running, a message indicates that "the game will launch into an environment in the HMD". However, no game is launched in the HMD.

If I instead choose "force the use of a compatibility tool" in Steam, and choose Proton 3.16-8 Beta, and launch the game while SteamVR is running, the game launches in VR, and is visible in the headset.
However, while in this mode, the VR wands are not visible, button/trigger presses have no effect, and pointing the wand in the direction of an on-screen button does not highlight it.
Also, no mouse cursor is visible, however, moving the mouse will highlight buttons, and clicking will click the buttons.

The wands do show up correctly in the SteamVR environment, and even in other games running under Proton, this problem seems to be unique to Tabletop Simulator.