Code sessions don't sync to mobile Remote Control (chat/cowork do)

Remote Control shows zero Code sessions on the mobile app, while chat and cowork sessions sync fine under the same account.

- Desktop: `remoteControlAtStartup: true` set in `settings.json`, multiple active Code sessions running (CLI v2.1.220)
- `.claude.json` cached GrowthBook flag: `cachedGrowthBookFeatures.tengu_onyx_plover` = `{enabled: false, minHours: 24, minSessions: 3, remoteEnabled: false}` — looks like an eligibility/rollout gate specific to Code session remote control
- Cleared the cached flag and restarted the app to force a re-fetch; server returned `false` again — so this is server-side, not a stale local cache
- `remoteControlSurfacesSeen: ["mobile"]` and `hasRemoteEnvironment: true` are both present, confirming mobile pairing itself works

Expected: Code sessions appear in mobile Remote Control same as chat/cowork. Actual: they never appear.

Reproduce the issue with remoteControlAtStartup enabled, active Code sessions, and a paired mobile Remote Control surface. Compare the cachedGrowthBookFeatures.tengu_onyx_plover value in .claude.json with the server response, then trace how Code sessions are filtered from mobile sync; done means Code sessions appear alongside chat and cowork sessions.

Written by the indexing model from the issue text.