grammar - "Name and I" or "name and me" when they are neither the object nor subject?

There have been many questions on this exchange about when to use phrases such as "John and I" vs. "John and me". The answer seems to be you that you use "John and I" when they are the subject of the verb and "John and me" when they are the object of the verb. However, sometimes it isn't so clear. For example, consider the following sentence

"A collaboration, between John and I, would lead to
close friendship"

In this sentence, I think the subject is the actual collaboration (not John and I) and the object is the friendship. So what is the proper usage in this case, where "John and I" is neither the subject nor the object?

Obviously this question is not a duplicate of any question asking about the use of "you" and "I" that doesn't contain rules for special prepositions such as "between".