Completeness of the cohomology obstruction for strong contextuality

A measurement scenario consists of a set of measurements together with a family of contexts, and a probabilistic empirical model assigns compatible probability distributions to the contexts. A model is strongly contextual when no global assignment of outcomes is compatible with its support, and its cohomology obstruction is the obstruction obtained from the associated cohomological construction. Completeness conjecture. Under suitable assumptions of symmetry and connectedness, the cohomology obstruction is a complete invariant for strong contextuality. The cohomological condition is known to be sufficient but not necessary in general; this conjecture proposes that symmetry and connectedness eliminate the false positives and make the obstruction characterize strong contextuality exactly.