Resolve the full Non-Cancelling Intersections conjecture

Determine whether the Non-Cancelling Intersections conjecture remains true in its full generality, without restricting witnesses to left-linear trees.

Background

The Non-Cancelling Intersections conjecture asserts that unions of finite families of sets can be constructively represented using algebraically non-cancelling intersections, disjoint unions, and subset complements. The paper disproves only the proposed left-linear-tree strengthening by constructing a lattice whose toggle game is unwinnable.

The unrestricted conjecture is not settled by that counterexample, because the failure of left-linear representations does not exclude representations using arbitrary dot-algebra parenthesizations. The authors explicitly identify determining whether their argument can be extended to refute the full conjecture as an unresolved question.