Validity of the conjectured cup-product formulas

Determine whether Conjecture 4.7 holds for the rational cohomology ring of the real toric space associated with an arbitrary Bier sphere Bier(K), including the stated formulas for cup products when the cohomology-generator degrees have different parity or when both degrees are odd and the indexing subsets intersect in exactly one element.

Background

The paper identifies rational cohomology generators of the real toric space M^R(Bier(K)) with subsets I belonging to the collections I_i(K). Theorem 4.5 establishes several cases of the cup-product structure, including products that vanish under specified intersection conditions and products of generators of even degree whose indexing subsets are disjoint.

Conjecture 4.7 proposes the remaining cup-product formulas: when the degrees have different parity and the indexing subsets are disjoint, the product should equal the generator indexed by their union up to sign; when both degrees are odd and the indexing subsets intersect in one element, the product should be determined by their symmetric difference, again up to sign, or vanish when that symmetric difference is not an admissible index. The authors report verification in small dimensions but explicitly state that the general validity remains unresolved. A proof would provide a full description of the multiplicative structure of the rational cohomology ring.