Feature request: Sync sessions across multiple VSCode Remote clients

Problem

When using Claude Code via VSCode Remote SSH from multiple client machines (e.g., work computer and home computer) connecting to the same remote server, session history is stored per-client rather than centrally.

• Deleting sessions on Client A does not reflect on Client B
• Each client has its own independent session list, even though they connect to the same remote server
• There is no way to keep session state consistent across clients

Expected behavior

Sessions should be synced across all clients connecting to the same remote server, so that deleting a session from any client removes it everywhere, and session history is consistent regardless of which client machine is used.

Use case

Users who work from multiple machines (e.g., office desktop and home laptop) via VSCode Remote SSH to the same dev server expect a unified session view, since all work happens on the same remote environment.

Possible approaches

• Store session metadata on the remote server side (e.g., ~/.claude/) rather than the local VSCode client
• Or provide a sync mechanism between clients