[FEATURE] Suggested-task chips should persist across app restarts (task inbox instead of ephemeral popups)

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet (related: #81213 covers the "Start with worktree" default-button regression, #75622 asks for ticket creation from suggested tasks — this request is about chip durability)
• [x] This is a single feature request (not multiple features)

Problem Statement

spawn_task chips ("Suggested task" cards in the Desktop app) are the best mechanism for turning discovered side-work — a bug found in passing, a follow-up task, the next step of a piece of work — into a one-click new session. But they're ephemeral: they don't survive an app restart, and they're easy to miss if you step away and come back hours later.

That makes them unreliable as the primary channel for "work the model queued up for you", so users have to duplicate every chip's content into the reply text as the durable record — which defeats the purpose of having a structured mechanism.

Proposed Solution

Persist pending chips and show them in a small task inbox (a badge/panel listing pending suggested tasks across sessions), surviving app restarts, until the user starts or dismisses them. The existing dismiss_task semantics (model can withdraw stale chips) would keep the inbox clean.

Alternative Solutions

Currently: every chip is mirrored as a line of reply text so at least the suggestion survives, and re-spawned manually after a restart if still relevant. Both are convention, not guarantee.

Priority

Medium - Would be very helpful

Feature Category

Other (Desktop app)

Use Case Example

1. During release QA, the model discovers a latent parsing crash — out of scope for the release — and spawns a chip "File bug ticket for latent contract-parsing crash".
2. I'm in meetings the rest of the day; the app restarts overnight (update).
3. Today the chip would be gone — the bug only gets filed if I happen to re-read yesterday's transcript. With a persistent task inbox, it's still waiting with a badge the next morning.