Automatically run awesome-lint in CI?

Some recent PRs mentioned tidying the markdown code with https://github.com/sindresorhus/awesome-lint -- is it something that we could/should do automatically during CI, similar to how we do pylint & black?

Review the existing CI checks for pylint and black, then inspect the linked awesome-lint project and its requirements. Determine whether it can run against this repository's Markdown; done means the CI configuration runs the lint check automatically and reports failures on pull requests.

Written by the indexing model from the issue text.