fix(host-control-mode): reconcile desync state on content reinjection

Fixes a split-brain where a guest who chose "watch on my own" and then reloaded
the video page would appear stuck as "Solo" to the host while actually being
back in sync.

Root cause: background persists hcmDesynced (survives SW restart) and attaches
it to content-driven heartbeats, but a freshly injected content script started
with hcmDesynced=false and GET_CONTROL_MODE never returned the desync state — so
content re-applied host commands (synced) while background kept relaying
desynced=true to the host.

Fix: GET_CONTROL_MODE now also returns desynced; content adopts it on init
(re-shows the badge) and also adopts hostPeerId so later host-change resets work.
Desync now survives a page reload consistently on both sides, as intended.

Verified the full path: background.js heartbeat (1555 omits / 2018 attaches),
reset paths, CONTENT_BOOT and keepAlive-disconnect (neither reset it), content
sendHeartbeat on inject.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

extension/content.js

@@ -1435,6 +1435,13 @@
         if (chrome.runtime.lastError || !res) return;
         hcmControlMode = res.controlMode || 'everyone';
         hcmAmHost = !!res.amHost;
+        hcmHostPeerId = res.hostPeerId || null;
+        // Re-adopt persisted desync after a page reload so we don't start synced
+        // while background still relays us as "Solo" to the host (split-brain).
+        if (res.desynced && !hcmDesynced) {
+            hcmDesynced = true;
+            hcmShowBadge();
+        }
     });
 
 })();