Claude in Chrome extension never connects (tabs_context_mcp always returns "not connected") despite reinstall + full browser restart

### Description:
The Claude in Chrome extension never establishes a connection with Claude Code. Every call to the browser tools returns:

▎ Browser extension is not connected. Please ensure the Claude browser extension is installed and running, and that you are logged into claude.ai with the same account as Claude Code.

### Environment:
- OS: Windows 11 Home
- Extension: Claude in Chrome, version
- Extension ID: fcoeoabgfenejglbffodgkkbkcdhcgfn
- Claude Code: CLI session on Windows

### Troubleshooting already attempted (all unsuccessful):
1. Confirmed the extension is installed and enabled (toggle "On") in chrome://extensions.
2. Confirmed logged into claude.ai in the same Chrome browser/profile as Claude Code.
3. Fully restarted Chrome (all windows closed, not just the tab) — multiple times.
4. Cleared the extension's service worker storage/data via chrome://extensions.
5. Fully uninstalled the extension and reinstalled it from the Chrome Web Store, then restarted Chrome again.

Start by reproducing the failure from a Claude Code CLI session with the Claude in Chrome extension installed, then call tabs_context_mcp and the browser tools. Check whether the extension remains connected after the documented Windows and Chrome setup steps. Done means the browser tools no longer return “not connected” and can access the browser session.

Written by the indexing model from the issue text.