GitHub authentication repeatedly loops in github.dev after opening a repository with the "." key

Does this issue occur when all extensions are disabled?: Not tested

* VS Code Version: github.dev web-based editor
* OS Version: Windows 11 25H2
* Browser: Google Chrome 150.0.7871.187
* GitHub account type: Personal account
* Repository visibility: Private

Steps to Reproduce:

1. Sign in to GitHub in Google Chrome.
2. Open a private repository on `github.com`.
3. Press the `.` key to open the repository in the `github.dev` web-based editor.
4. Wait for the editor to load.
5. Complete the GitHub authentication process when prompted.
6. Return to the editor.
7. The authentication prompt appears again.
8. Authentication repeatedly loops, and the repository cannot be accessed normally.

Expected behavior:

After completing authentication, the repository should open in `github.dev` and remain signed in.

Actual behavior:

The GitHub authentication process repeatedly restarts, resulting in an infinite authentication loop. Because of this, I cannot access or edit the repository in `github.dev`.

Additional information:

This previously worked correctly in the same browser and environment, but the issue started occurring suddenly.

I contacted support about this issue and was advised to report it in the `microsoft/vscode` repository.

I previously submitted a HAR file through the support request, but I no longer have a local copy of it. I have not attached a new HAR file to this public issue because HAR files may contain authentication-related or personally identifiable information.

I can collect additional diagnostic information if a maintainer specifies what information is required and how it should be shared safely.

The Insiders build and `code --disable-extensions` instructions do not appear to apply because this issue occurs in the `github.dev` web-based editor.