Finite convergence and the \(\alpha(G)\geq 9\) case for \(\nu^{(r)}\)

Determine whether, for graphs with stability number \(\alpha(G)\geq 9\), the equality \(\nu^{(\alpha(G)-1)}(G)=\alpha(G)\) holds and whether the hierarchy \(\nu^{(r)}(G)\) has finite convergence for every graph.

Background

The hierarchy $\nu^{(r)}(G)$ is based on the more restrictive SOS cones $Q_n^{(r)}$. The equality at level $\alpha(G)-1$ is known for graphs with $\alpha(G)\leq 8$, but the paper explicitly identifies the extension to larger stability numbers as unresolved. It also distinguishes this issue from the corresponding $\vartheta$-hierarchy, for which finite convergence had recently been proved, and states that finite convergence of $\nu^{(r)}$ itself was not known.