Claude-in-Chrome extension not connecting from cloud/Cowork session despite extension active locally

In a Cowork cloud session, tabs_context_mcp (and other mcp__claude-in-chrome__* calls) consistently return "Browser extension is not connected," across roughly 6 retries over ~30 minutes, including after restarting the session entirely. The user confirms the Claude extension is installed, shows as active in Chrome (tab group visible), and they're logged into claude.ai. The issue persisted across a fresh session, suggesting it's not a one-off state glitch but a connection/account-linking problem between the cloud session and the local extension.

Start by reproducing tabs_context_mcp and other mcp__claude-in-chrome__* calls in a fresh Cowork cloud session while the Claude extension is active locally. Trace the cloud-session and local-extension connection or account-linking path; done means the calls connect reliably without the persistent “Browser extension is not connected” error.

Written by the indexing model from the issue text.