The integration tests workflow is broken

I have a fork of this repository, and every week I get an e-mail about a build failure due to the `integration_test.yaml` workflow. I checked on the original repo, and apparently that workflow is manually disabled. In my repo, this workflow fails due to broken links (but these links appear to work fine, I suspect the runner is firewalled or something). I see this was added in #3645 and #3658, but I couldn't find any issue or PR about its problem or disabling it. Can we fix this workflow or remove it?

Start with integration_test.yaml and review the workflow changes introduced in #3645 and #3658. Reproduce the broken-link failure in a fork or inspect the runner and network behavior; done means the workflow passes reliably or its removal is clearly chosen and implemented.

Written by the indexing model from the issue text.