claude-in-chrome: extension shows connected/logged-in but tools always report 'Browser extension is not connected'

Description

All mcp__claude-in-chrome__* tool calls (tabs_context_mcp, navigate, etc.) consistently return:

Browser extension is not connected. Please ensure the Claude browser extension is installed and running (https://claude.ai/chrome), and that you are logged into claude.ai with the same account as Claude Code.

...even after extensive troubleshooting confirmed the extension itself is fine.

Environment

• OS: Windows 11 Pro 10.0.26200
• Chrome version: 150.0.7871.187
• Claude Code running via CLI on Windows (PowerShell/Git Bash)

Troubleshooting already performed (all failed to resolve)

1. Fully closed and reopened Chrome
2. Reinstalled the "Claude for Chrome" extension from https://claude.ai/chrome
3. Verified in chrome://extensions that the extension is enabled, with no error badge shown
4. Clicked the extension toolbar icon and confirmed it shows as logged in with the same claude.ai account used for Claude Code (gooline@gmail.com)
5. Closed all other open browser tabs to rule out resource/tab-count issues
6. Retried the tool call ~7 times across all the above states — every single attempt returned the identical "not connected" error, with no variation

Expected behavior

With the extension installed, enabled, and logged into the same account as the CLI session, tabs_context_mcp (or any other claude-in-chrome tool) should establish a connection successfully.

Actual behavior

Connection never succeeds; error message is identical regardless of extension/login state changes on the Chrome side, which suggests the issue may be in the native-messaging/relay layer between the CLI and the extension rather than anything user-configurable in Chrome itself.