Feature request: shared/exportable conversation history between Claude desktop app and Claude Code

Is your feature request related to a problem? Please describe.
I use both the Claude desktop app (macOS) and Claude Code on the same workstation, and there's currently no way to share context between them — each maintains a completely separate conversation history with no sync, import, or export path. This creates friction when switching between exploratory/chat work in the desktop app and hands-on coding work in Claude Code on the same task.

Describe the solution you'd like
One of the following, in order of preference:

1. Export/Import: The ability to export a conversation (or its key context) from the desktop app and import it into a Claude Code session, and vice versa.
2. Shared history/memory layer: An opt-in way for Claude Code and the desktop app to reference a common conversation or memory store when running on the same machine/account, so context established in one carries into the other.
3. Manual bridge (minimum viable): A simple way to copy a desktop app conversation as structured text/markdown that Claude Code could ingest as context (e.g., via a slash command or file drop).

Describe alternatives you've considered
Manually copy/pasting relevant context between the two products — works but is tedious and easy to lose track of, especially across longer sessions.

Additional context
Both products run under the same account on the same machine, so even a basic same-device bridge (not full cloud sync) would solve most of the friction.