Vscode auth flow: cannot redirect to correct instance of vscode

I open multiple instances of vscode in ubuntu by specifying user-data-dir, for example: code --user-data-dir=~/.config/vscode_white,

Now in each instance, I try to log in to GitHub Copilot, but after authentication, it only redirects to the default vscode instance running in user-data-dir=~/.config/vscode

How can I redirect to the correct vscode instance running in user-data-dir=~/.config/vscode_white?

This is the sample login URL, browser will ask to open vscode application

https://github.com/login/oauth/authorize?client_id=12321313123213131&redirect_uri=https://vscode.dev/redirect&scope=user:email&skip_account_picker=true&state=vscode%3A%2F%2Fvscode.github-authentication%2Fdid-authenticate%3Fnonce%3D12312313123213%26windowId%3D1234

I suspect we need to update this URL to redirect to the correct vscode instance, I tried to update windowId to the value got from this command xdotool search --onlyvisible --pid $PID but noluck