On the ongoing experiments looking for higher-order interference: What are they really testing?

The existence of higher than pairwise quantum interference in the set-up, in which there are more than two slits, is currently under experimental investigation. However, it is still unclear what the confirmation of existence of such interference would mean for quantum theory -- whether that usual quantum mechanics is merely a limiting case of some more general theory or whether that some assumption of quantum theory taken as a fundamental one does not actually hold true. The present paper tries to understand why quantum theory is limited only to a certain kind of interference.