"Open Claude Code" handoff from claude.ai doesn't carry conversation context

The "Keep going in Claude Code" button on claude.ai (with copy "Switch to Claude Code and let Claude work directly in your repo, running and testing as it goes.") implies the conversation will continue. In practice, it just opens a fresh CLI session in a directory — no transcript, no attached screenshots, no working context from the web chat carries over.

This is misleading. Either:

• the handoff should actually transfer a summary/context as the first message in the new session, or
• the button copy should make it clear that you're starting fresh (e.g. "Open Claude Code in this project" rather than "Keep going").

As written, users are surprised to find the new session has no memory of what they were just discussing.