Currently, nogil build has been added to buildbot, but it is inconvenient to check whether it is working properly in the PR stage.
So I would like to suggest: Putting a CI that checks nogil builds in Github Action.

This will satisfy the following conditions:

• It will only run against the main branch
• It will check only the build itself and does not run unittests
• (Optional) Even if it fails, it does not fail and does not interfere with other PRs.
  (e.g gh-263: Run Python 3.12 task as the experimental task pyperformance#274)

If this happens, it will be very helpful in reviewing nogil-related PRs.

Linked PRs

• gh-111062: Add nogil build test on GitHub Actions #111060
• [WIP] gh-111062: Add free-threaded CI for Windows(x64) #111437
• GH-111062: Apply reusable workflow for the Linux CI #111440
• gh-111062: Separate macOS build into a reusable workflow #111444
• gh-111062: Separate linux build into a reusable workflow #111446
• gh-111062: Build both default and free-threaded on macOS #111449
• gh-111062: Reusable Ubuntu build that supports free-threaded mode as the conditional CI #111452
• gh-111062: Reusable Windows build that supports free-threaded mode as the conditional CI #111493
• gh-111062: Update name of reusable workflows for Ubuntu / macOS #111494
• gh-111062 CI: Update skip lists for resuable workflows #111521
• gh-111062: CI: Rename reusable-build-[X].yml to reusable-[X].yml #111552
• [3.12] gh-111062: CI: Move OS test jobs to reusable workflows #111568
• [3.11] gh-111062: CI: Move OS test jobs to reusable workflows #111570