hyphenation - Mixing hyphenated prefixes with cased hyphenated compound modifiers

Suppose I know two professors. Both of them are old, and both of them study English. Then each one is an old English professor.

Suppose one and only one of them is a scholar of Old English. Then that one would be an Old English professor.

So far this is unambiguous, according to https://en.wikipedia.org/wiki/English_compound#Hyphenated_compound_modifiers

But now I have a question about the other professor.

Is he a non-Old English professor or a non-Old-English professor?