Fullscreen windows don't have a fullscreen window hint

Problem

I'm using Awesome tiling window manager, which supports Extended Window Manager Hints and relies on them to determine whether to treat clients as fullscreen or as regular tiled or floating windows. For some reason, games that run through Steam Play and use exclusive fullscreen don't have a fullscreen hint while windows that use borderless fullscreen do. See two screenshots below in case of Yakuza 0 with improvised debug notifications. First is with Window Mode set to Fullscreen (notice how lower (newer) notifications say "Fullscreen: false") and second is with it set to Borderless (notice how notifications say "Fullscreen: true").

Affected games

Potentially all games with exclusive fullscreen. Tested on:
#2309 (no borderless option)
#492
Need for Speed Heat

Current workaround

Attach toggling fullscreen to a keybind (default is Mod4 (Super) + F). Or do some trickery in the configs

Setup

OS: Arch Linux
Kernel: linux-zen 5.14.12-zen1-1-zen
WM: awesome v4.3-1220-g1d55ae09a
Compositing manager: picom-jonaburg-git vgit-a8445
GPU drivers: nvidia-dkms 470.74-1

Reproduce the issue with an affected Steam Play game in exclusive fullscreen and compare its Extended Window Manager Hints with borderless fullscreen under Awesome. Start by tracing Proton's fullscreen-window handling; done means exclusive-fullscreen windows expose the expected fullscreen hint without relying on the keybind workaround.