spawn_task chips should persist across app restarts

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

Task chips created via spawn_task (the background task suggestion UI in the upper-right corner) are lost when the Claude Code app is closed and reopened. This makes them effectively useless for any task that spans multiple sessions, which is their primary use case.

Current behavior

When Claude pins a suggested follow-up task using spawn_task, the chip appears in the session UI. Closing and reopening the app removes it with no recovery path. The spawn_task documentation acknowledges this: "Task ids are not persisted across app restarts."

Proposed Solution

Expected behavior

Pending (unacted-upon) task chips should survive app restarts, at minimum until the user explicitly dismisses them or acts on them.

Suggested fix

Persist pending task chip state (id, title, tldr, prompt) to local storage or the settings file, and restore unacted chips on next launch.

Alternative Solutions

User Workaround

Ask Claude to print the raw prompt text so it can be saved externally.

Priority

Low - Nice to have

Feature Category

Interactive mode (TUI)

Use Case Example

Why this matters

The spawn_task tool is designed for out-of-scope items flagged during a session — things like "noticed a bug, don't fix it now, do it later." That workflow is inherently cross-session. A chip that evaporates on close requires the user to manually copy the prompt text somewhere else, defeating the purpose of the chip entirely.

Additional Context

_No response_