feat(desktop): surface parent task status in Side Conversations

### Problem

Focused child of #4331: while consulting a Desktop Side Conversation, users need to see when the parent task requires input or approval, fails, is interrupted, or completes. The companion panel currently shows its own transcript, execution indicator and interactions; it does not provide an authoritative parent-status notice.

Verified by code inspection on main `83aa12a29`. A Session summary alone is insufficient: `active` is not proof of completion, and the Desktop display projection normalizes some failure states. The notice must use existing Host-backed execution/interaction facts and settled Turn records.

### Desired outcome

Add a compact, localized parent-status notice to the existing Side Conversation panel:

- Distinguish parent input/approval, running, last-turn completion, failure and interruption. Label terminal information as the latest parent turn, including when opening the panel after that turn ended; do not report a new/idle Session as completed.
- Reuse canonical Host-backed execution and pending-interaction data. Audit approval coverage rather than silently interpreting only one kind of approval as complete coverage.
- Provide a route to the existing parent conversation for actionable states; do not introduce another approval handler.
- Refresh correctly across interaction resolution, new turns and reconnect. Show unavailable/unknown when authoritative state cannot be read, and reject stale replies after navigation or disposal.
- Keep the companion draft, transcript and execution unchanged. Avoid duplicate subscriptions where an existing owner already provides the required facts.

Acceptance coverage: all parent states; pending input/approval becoming resolved; failure versus idle/completed; reconnect hydration; source Session switch and delayed replies; unmount cleanup; visible localized notice and parent navigation with keyboard access.

This claims only the parent-status item. It does not cover tab descriptors (#4744), linked-session retention (#4494 / #4625), model selection, durable promotion, or Runtime permission/execution policy changes.

### Alternatives or workarounds

Switch back to the parent conversation and inspect it manually. Reusing only the catalog status is simpler but cannot truthfully distinguish the required outcomes.

中文摘要：在侧边对话中显示主任务的等待输入、等待审批及最近一轮完成/失败/中断状态，沿用真实状态来源和主对话处理入口。仅认领 #4331 的这一子项，不占用其他贡献者范围。

Prepared and posted by OpenAI Codex on behalf of @hbw00111, who authorized this contribution. Implementation will use Grok Build with Codex verification.