feat(vdisplay/gamescope): honor in-stream 'Switch to Desktop' under the managed DM takeover

First on-glass test of the DM-flavor takeover confirmed the black-screen
fix, but exposed that the managed model could not switch back: Steam's
in-stream session-select is a silent no-op while the display manager is
stopped (every config-write branch in os-session-select requires the DM
to be running), so the capture loss it causes just relaunched game mode.

The user pass of the switch script does leave one durable trace — the
~/.config/steamos-session-select sentinel, written before any DM check.
The managed launch now baselines its mtime; a capture loss with the
sentinel advanced reads as the switch request and is honored by
replaying the switch with the DM up: restore the display manager, run
the distro's own os-session-select (its internal pkexec is authorized
allow_any by the distro policy, so it works from the host's sessionless
context), then stop the autologin game-mode unit so Relogin lands in
the newly selected desktop. The capture-loss re-detection then follows
KWin as it comes up; a 120 s post-honor grace stops the rebuild loop
from racing the booting desktop back into game mode (superseded early
if the box's own game-mode unit reappears — the desktop→game leg stays
fast). The baseline is re-recorded on crash-restore so a pre-existing
sentinel never reads as a fresh request. Every verb in the sequence was
live-validated on the Nobara repro VM.

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>

docs-site/content/docs/gamescope.md

@@ -60,6 +60,10 @@ depends on the display manager driving the autologin:
   `PUNKTFUNK_RECOVER_SESSION_CMD` (see [Configuration](/docs/configuration)) is fired as the
   fallback.
 
+  With the rule in place the **in-stream session switch round-trips** in managed mode: Steam's
+  "Switch to Desktop" inside the streamed Game Mode returns the box to its desktop session and the
+  stream follows it there; the desktop's "Return to Gaming Mode" switches it forward again.
+
 ## Session following
 
 `PUNKTFUNK_SESSION_WATCH` follows a Gaming ↔ Desktop switch **mid-stream** — the host rebuilds the