Dimension-independent exactness level for the \(\tilde{Q}\) hierarchy

Determine whether there exists a function \(r(n)\) such that \(\tilde{Q}^{(r(n))}_n=\mathcal{C}_n\) for every dimension \(n\), thereby characterizing whether the exactness observed at dimension five extends to all dimensions.

Background

The paper proves that the proposed hierarchy captures the entire 5×55\times5 copositive cone at level one, yielding an SDP-feasibility characterization in that dimension. It explicitly leaves unresolved whether dimension five is exceptional or whether every dimension admits some finite exactness level.

References

Nonetheless, it remains an open question whether the case n=5 is exceptional for the {(r)} hierarchy, or if there exists a convergence pattern across all dimensions; is there a function r(n) such that {r(n)}_n = _n ?

— 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