Counterexamples for the hypermetric facets of the six-vertex cut cone

Determine whether the two new hypermetric facet inequalities required to describe the six-vertex cut cone yield a counterexample to the bounded-ratio conjecture asserting that every normalized reduced bounded ratio on the cone of positive Lorentzian matrices has value at most 2.

Background

The paper explains that the bounded-ratio conjecture is known for dimensions n≤5 and that the facets of the cut cone correspond to extreme rays of the cone of reduced bounded ratios on positive Lorentzian matrices. For the six-vertex cut cone, two additional facet inequalities of hypermetric type are needed beyond the inequalities inherited from five-vertex submatrices.

The authors construct a counterexample in dimension n=7 using a non-hypermetric clique-web facet, thereby disproving the general conjecture. However, they explicitly state that they were unable to produce a counterexample associated with the two new hypermetric facets for n=6, leaving unresolved whether those facets also violate the conjectured bound.

References

According to *{Remark 15.2.11} there are, up to symmetry and in addition to the inequalities on $5\times5$ submatrices, two new inequalities needed to describe $\Cut_6$. These are of a certain type that is called hypermetric and we were not able to produce a counter-example for them.

— A note on bounded ratios  (2609.03934 - Baldi et al., 3 Sep 2026) in Section 3, “A counterexample to the bounded ratio conjecture”