Convex cone hierarchy closed under scaling and bordering

Construct a hierarchy of convex cones that approximates the copositive cone and is closed under both diagonal scaling and bordering, or determine whether any intermediate cone with all three properties exists.

Background

The proposed hierarchy Q~n(r)\tilde{Q}_n^{(r)} is closed under borderings and diagonal scalings, but its levels are generally nonconvex. The paper notes that the level-zero cones possess convexity together with these structural closure properties, and asks whether an intermediate convex hierarchy can retain the same structure while approximating the copositive cone.

References

It is natural to consider the existence of a hierarchy of convex cones that maintains closure under both scaling and borderings, and that approximates the copositive cone. Notably, the cones Q{(0)} and $$ exhibit all three desired properties. Nonetheless, it remains an open question as to whether there exists even an intermediate cone that satisfies these criteria.

— Low degree sum-of-squares bounds for the stability number: a copositive approach  (2509.04949 - Vargas et al., 5 Sep 2025) in Section 10, Conclusion and Discussion