[DO NOT MERGE] experiment: test auto-formatting in CI for internal PRs

[pepopowitz]

This is test #1 for #1130 -- internal PRs.

With #1130 configured, if I push up a PR for a branch within this repo, does the GITHUB_TOKEN running the auto-format workflow have write access to the branch?

Findings

We can see in the commit log that the workflow did commit a formatting change (note the 2nd commit):

Looking at the output from the workflow, we can see the commit happening successfully:

Surprisingly, the next step (build-docs) failed, on verifying the formatting. It looks like it checked out the commit at the beginning of the previous step, before the fix was committed.

I'm pretty sure we'd be able to resolve this by updating the build-docs workflow, but I'm not going to look into it because test #2 for #1130 failed.

[pepopowitz]

Closing this PR as it is an experiment, and was only created for posterity.