claude-in-chrome: WebSocket disconnects during long sessions, no auto-reconnect

Description

The claude-in-chrome MCP connection drops intermittently during long Claude Code sessions (>30 minutes). Once disconnected, tabs_context_mcp returns "Browser extension is not connected" and there is no way to re-establish the connection without restarting both Claude Code and Chrome.

Steps to Reproduce

1. Start a Claude Code session with the Chrome extension connected
2. Use mcp__claude-in-chrome__tabs_context_mcp — works fine
3. Continue working for 30-60 minutes (mix of Chrome tool calls and non-Chrome work)
4. Call mcp__claude-in-chrome__tabs_context_mcp again — returns "Browser extension is not connected"
5. The Chrome extension is still running and visible in Chrome — it's the MCP WebSocket that dropped

Expected Behavior

• The MCP transport layer should detect the dropped WebSocket and auto-reconnect with exponential backoff
• If reconnection fails after N retries, surface a clear error with a manual reconnect option
• Tab group context should be recoverable after reconnection

Actual Behavior

• Connection silently drops with no retry
• All subsequent mcp__claude-in-chrome__* calls fail
• Only fix is to quit Claude Code and restart Chrome
• This happens consistently in sessions longer than ~30 minutes

Impact

For workflows that depend on browser automation (reading/writing page state, posting to localStorage, navigating), this forces frequent session restarts and breaks long-running tasks. The user has to manually restart both applications each time.

Environment

• Claude Code: latest (claude-opus-4-6)
• Chrome: latest stable, Linux (Ubuntu 24.04)
• The Chrome extension is installed, logged in, and shows as active in Chrome

Suggested Fix

Standard WebSocket reconnection with exponential backoff:

1. Detect connection drop (heartbeat/ping-pong or error handler)
2. Retry with backoff: 1s, 2s, 4s, 8s, 16s, max 30s
3. Re-establish tab group context after reconnection
4. Surface connection status to the user (e.g., "Reconnecting to Chrome...")