Since I couldn't find any reason why Docker is keeping this cache, I used the following workaround in case other users are facing the same issue with the same type of project.

I stopped using the management command and the dedicated app cron container.

I moved those management commands to REST views that only a specific user with a specific auth token can launch.

I created a new simple container for a cron job that performs curl requests on those URLs.

Everything works now.