Add comprehensive Playwright integration tests for chat unread badge:
- Badge visibility based on count (hidden at 0, visible with count)
- Badge text formatting (1-99, "99+" for 100+)
- Badge reset when window opened
- Badge increment logic (closed window vs different channel)
- Multiple message handling
- Badge state after page reload
Tests validate Redux state, localStorage persistence, and UI rendering.
Related to Phase 10 Plan 1: Read/Unread Status Validation & Testing
Co-Authored-By: Claude Sonnet 4.5 <noreply@anthropic.com>
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