Counterexamples with all three exact-one complements sufficiently dense

Determine whether there exists a counterexample consisting of three upward closed systems X, Y, and Z for which each of the three sets Xᶜ∩Yᶜ∩Zᶜ, Xᶜ∩Y∩Z, and X∩Yᶜ∩Zᶜ has density greater than 27/2⁷ ≈ 0.148, and determine whether their densities can each exceed 0.153.

Background

The paper studies the density of points contained in exactly one of three upward closed systems on the Boolean lattice. It establishes an upper bound of approximately 0.515 and constructs a counterexample with density approximately 0.447, leaving a gap between the known lower and upper bounds.

The conclusion records questions raised by Bhargav Narayanan, Jeff Kahn, and Sophie Spirkl concerning whether counterexamples can be strengthened so that each of three specified membership-pattern classes has density above specified thresholds. The paper explicitly states that these questions remain open.