de Klerk–Pasechnik finite-convergence conjecture

Prove that for every graph G, the sum-of-squares hierarchy \(\vartheta^{(r)}(G)\) reaches the stability number at level \(\alpha(G)-1\), namely, establish \(\vartheta^{(\alpha(G)-1)}(G)=\alpha(G)\).

Background

The paper studies the hierarchy $\vartheta^{(r)}(G)$ obtained by replacing the copositive cone in the copositive formulation of the stability number with the SOS-based cones $K_n^{(r)}$. De Klerk and Pasechnik conjectured that this hierarchy is exact after $\alpha(G)-1$ levels. Although finite convergence has since been established, the paper states that no general bound on the number of levels required is known, so the conjectured bound remains unresolved.