Get or Check IE compatibility view settings

Question

I am working on a single sign on between an page I control and a page I do not. Unfortunately, their page needs to be viewed in compatibility view if you are using IE9 or higher. I have built out a function to check for

if(!document.documentMode || document.documentMode < 8)

before redirecting, but sadly that will only check if my page is in the compatibility view list, what I need to do is check if their page is in the list. Is this possible in Java, JavaScript, or JSP, and if so, how? I have looked online and no one makes any mention of checking if another site will be viewed in compatibility view.

Note: The provider of the site I am connecting to refuses to make any adjustments for this purpose.