Add per-pane new thread draft slots

[brsbl]

What was wrong

New-thread compose state was keyed as one browser-wide prompt draft. A second pane therefore reused the same draft identity, and the app could not preserve independent unsent composers or their creation destinations across navigation and layout restoration.

What changed

• Adds client-local draft slots with stable IDs, timestamps, and project/section submit-destination metadata.
• Keys every app-owned new-thread pane and route to an explicit slot while keeping plugin-rendered composers outside Drafts state.
• Opens a fresh empty slot for every ordinary route into a new-thread composer; only explicit draft/pane re-entry reuses a slot.
• Migrates legacy singleton prompt storage and split-layout persistence deterministically, including corrupt/empty-state cleanup.
• Preserves slot identity through split navigation, focus, maximize, close, remount, and sidebar modifier/drag interactions.
• Adds no server state, server files, wire-contract changes, or daemon protocol changes.

Before — Phase 1 parent (c781d323e)

The same modified New thread action remains a singleton composer.

After — this PR (e503e4d35)

Two panes retain independent composer text in separate persisted draft slots.

How you verified

• Focused draft-slot, compose-route, split-layout, sidebar split-drag, and plugin-composer regressions: 11 files / 160 tests passed.
• pnpm exec turbo run typecheck: passed under Node 22.22.1.
• pnpm exec turbo run lint: passed under Node 22.22.1.
• Full @bb/app test suite: 436 files; 3,370 passed and 3 skipped.
• Full repository test graph reached 69/75 tasks; the only failure was the unchanged bb-plugin-provider-claude-code recorded-conformance replay harness, which reported different missing replay cells across retries. This branch has no diff in that package.
• Chrome for Testing 151.0.7922.71: verified initial paint, hard reload/restoration, modified New thread split creation, two independently editable/persisted composers, and right-panel collapse at 1440×900. Safari verification is still pending and this PR is not presented as merge-ready.

BB-Thread-ID: thr_2e2gbjx943

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