Share conversation history between claude.ai and Claude Code for the same account

Claude Code (VS Code / CLI) and claude.ai are completely separate products with no shared memory or conversation history, even when the user is logged into the same Anthropic account. This forces users to re-establish context every time they switch between interfaces, and work done in one session is invisible to the other.

Request: Allow Claude Code to access claude.ai conversation history (or a shared memory store) tied to the authenticated account, so context is not lost when switching between products. At minimum, allow users to export a claude.ai conversation in a format Claude Code can import.