Can a consistent theory beyond quantum mechanics have three-path interference?
In plain words
Quantum theory forbids interference that needs three paths at once. The question is whether any other self-consistent probabilistic theory allows it, which fixes what a positive three-slit test would mean.
Precise statement
Within generalized probabilistic theories (GPTs), determine whether a theory with nonzero third-order interference $I_3$ exists that satisfies causality, tomographic locality and a consistent composition rule with entangled states. Barnum, Lee, Scandolo and Selby (Entropy 2017, arXiv:1704.05106) showed that causality, purity preservation, pure sharpness and purification together exclude higher-order interference; the task is to find which postulates can be dropped.
What would settle it
An explicit GPT with $I_{3} != 0$ obeying the stated postulates, or an exclusion theorem from weaker postulates.
Status in the literature
Excluded under purity principles in 2017 (arXiv:1704.05106); which postulate must fail to allow higher-order interference is open.