[BUG] Desktop app session list does not show sessions created in VS Code extension

Description

Sessions created in the VS Code extension are visible in the VS Code extension's session sidebar, but the same sessions do not appear when using the desktop app (Mac). The sync appears to be one-directional — VS Code shows all sessions from both clients, but the desktop app only shows its own.

Steps to reproduce

1. Start a session in the Claude Code VS Code extension (same account, same project directory)
2. Open the Claude Code desktop app to the same project directory
3. Run session list — sessions from the VS Code extension are missing

Expected behavior

All sessions for the same account and project directory should be visible across all clients (VS Code extension, desktop app, web).

Actual behavior

Desktop app session list returns empty / only shows sessions started within the desktop app itself.

Environment

• macOS 25.5.0
• Same Anthropic account across both clients
• Same working directory (/Users/user/repo)

Impact

Users cannot continue VS Code extension sessions from the desktop app, breaking cross-client continuity.