Sessions created on Mac desktop app don't sync to mobile app session list (even after 'nudge' message and full app restart)

## Summary
Sessions created/run on the Mac desktop app (Claude Code) intermittently fail to appear in the mobile app's session list, even after triggering activity on the session and fully restarting the mobile app.

## What we observed (2026-09-12)

**Case 1 — regular interactive session, not visible on mobile:**
- A normal session was created and actively used on the Mac desktop app (title: 「奥さん用ホームページ作成」, `isRemote: false`, not a scheduled task).
- It did not appear in the mobile app's session list.
- We sent it a short message via the session-messaging tool to try to trigger re-sync (this had worked for a similar issue the day before, 2026-09-11, when several sessions were created near-simultaneously and some failed to sync). The message was delivered and the session's `lastActivityAt` updated (confirmed via the session API), meaning the session was live and reachable.
- Despite that, the session **still did not show up in the mobile app**, even after:
- Pull-to-refresh on the session list
- Fully quitting and reopening the mobile app
- Mobile app's session list (screenshot) showed only a single, unrelated session (the one currently active via Remote Control from that phone) under the "すべて" (All) filter — the missing session was nowhere in the list.
- Confirmed the phone is logged into the correct/same account, and Remote Control pairing with the Mac shows as active/paired.

**Case 2 — scheduled-task session, not visible on mobile (possibly expected):**
- A scheduled-task-dispatched session (a daily automation task) was also not visible on mobile.
- Confirmed via the user directly observing the Mac that this session was genuinely active (not stuck).
- Attempting to send it a message failed with: `session is unattended (a scheduled-task run or dispatched session); messages can't be delivered there.`
- This may be expected/by-design for unattended scheduled-task sessions (i.e., maybe they intentionally don't live-sync to the mobile session list until completion) — flagging mainly for context alongside Case 1, since both were reported as "invisible" by the user.

## Prior related occurrence (2026-09-11)
The day before, creating ~3 sessions on the Mac within seconds of each other resulted in 2 of them not appearing on other devices (phone, another laptop). At the time, sending a short message to the affected sessions successfully forced them to reappear on other devices. That workaround is what we tried again for Case 1 above, but this time it did not resolve the issue.

## Impact
User relies on starting/monitoring work from the Mac and checking in from mobile throughout the day (runs many parallel sessions for different small businesses). Sessions silently missing from the mobile list makes it hard to track or resume work while away from the Mac.

## Environment
- Desktop: Claude Code desktop app on macOS (iMac)
- Mobile: Claude Code mobile app (iOS), Remote Control paired with the Mac
- Model: claude-sonnet-5

Happy to provide session IDs / more detail via a private channel if useful — didn't want to post account-identifying details in a public issue.

No source file or test is identified. Start by reproducing the Mac-created session flow and inspect the session API, mobile session list, and session-messaging behavior; compare it with the documented scheduled-task restriction. Done means ordinary Mac sessions reliably appear on the paired iOS session list after creation and activity, with the scheduled-task behavior clarified.

索引モデルが issue の本文から書いたものです。