[BUG] Claude Code VSCode extension cannot persist in sidebar

Preflight Checklist

• [x] I have searched existing issues and this hasn't been reported yet
• [x] This is a single bug report (please file separate reports for different bugs)
• [x] I am using the latest version of Claude Code

What's Wrong?

Claude Code in sidebar position restarts the whole process once the sidebar is closed or switcher to another tab.
Better watch the video.
https://github.com/user-attachments/assets/44d1f8e4-cfbc-46c6-a396-7845326142bb

What Should Happen?

Claude Code must be persistent and do not stop the process once I switched to file explorer or another tab.

Error Messages/Logs

Steps to Reproduce

1. Open CC in any sidebar (I have tested left and right, no difference).
2. Start coding
3. Switch to another sidebar tab

Claude Model

None

Is this a regression?

No, this never worked

Last Working Version

_No response_

Claude Code Version

2.0.9 (Claude Code)

Platform

Anthropic API

Operating System

macOS

Terminal/Shell

Other

Additional Information

_No response_