Hide ContentType Field in Custom Document Set CT

Question

I have a Custom Document Set Content Type “Application Document Set”

In the edit mode at the very first row there is a field called “Content Type”, it is dropdown list and it is built in field. In my case I want to Hide it. I have tried programmatic and declarative way, both of them are not working. It is still showing in Edit Form.

Declarative:

<FieldRef ID=" c042a256-787d-4a6f-8a8a-cf6ab767f12d " Name="ContentType" DisplayName="Content Type" Hidden="TRUE" ShowInEditForm="FALSE"/>

Programmatic:

            SPList list = web.Lists["Applications"];
            SPContentType contentType = list.ContentTypes["Application Document Set"];
            //SPFieldLink link = contentType.FieldLinks[SPBuiltInFieldId.ContentType];
            SPField field = contentType.Fields[SPBuiltInFieldId.ContentType];
            field.Hidden = true;
            field.ShowInEditForm = false;
            contentType.Update();
            list.Update();
            web.Update();

Any idea? Thanks