GitHub Actions extension fails with "Unable to connect to the GitHub API" in .code-workspace (works with Open Folder)

## Description
When I open the same repository via a multi-root `.code-workspace`, the GitHub Actions extension shows:

> Unable to connect to the GitHub API, please check internet connection

If I open the repository directly with **File → Open Folder**, the extension works normally (workflows/runs load, authentication is fine).

This appears specific to workspace-mode repository resolution/activation, not network connectivity.

## Steps to reproduce
1. Create/open a multi-root workspace that includes this repo as a root:
- `c:\Users\\Source\loom-workbench` (plus other roots in workspace)
2. Open VS Code using the `.code-workspace` file.
3. Open the GitHub Actions view.
4. Observe error: **"Unable to connect to the GitHub API, please check internet connection"**.
5. Close VS Code.
6. Reopen only `c:\Users\\Source\loom-workbench` via **Open Folder**.
7. GitHub Actions extension works correctly.

## Expected behavior
The extension should authenticate and connect to GitHub API the same way in both:
- **Open Folder** mode
- **.code-workspace** (multi-root) mode

## Actual behavior
- **Open Folder**: Works
- **.code-workspace**: Fails with API connection error

## Environment
- Version: 1.125.1 (user setup)
- Commit: fcf604774b9f2674b473065736ee75077e256353
- Date: 2026-06-19T08:39:34Z
- Electron: 42.2.0
- ElectronBuildId: 14159160
- Chromium: 148.0.7778.97
- Node.js: 24.15.0
- V8: 14.8.178.14-electron.0
- OS: Windows_NT x64 10.0.26200

## Additional context
- Internet connectivity is fine.
- Authentication/session is valid (confirmed by successful behavior in Open Folder mode).
- Reproducible consistently with the same repository and machine.

Reproduce the failure using the reported multi-root .code-workspace and compare it with opening the same repository through Open Folder. Trace the extension's workspace-mode repository resolution and activation path; done means GitHub Actions workflows and runs load in both modes without the API connection error.

Written by the indexing model from the issue text.