AgileApps Cloud - removing the "Add record" button from subforms/related information


I have a list of Case objects whose forms allow users to add child records through subforms.
After creating and saving the Case, I’d like to make the entire form “read-only” for every user, also in the subform/related information sections. Thanks to layout rules, the operation is pretty easy on the Case object form, but I cannot prevent users to add new related records through the “Add record” button in the subform section.
Is there a way to do that? The goal is to make my Cases fully read-only after users create them, so nobody should be able to add new child record.

Thank you in advance.