Gvozdenović–Laurent bound for the copositive stability-number hierarchy

Let $G=(V,E)$ be a graph, let $\vartheta^{(r)}(G)$ be its order-$r$ copositive hierarchy bound, and let $S^\perp$ denote the vertices adjacent to or equal to a stable set $S$. For every stable set $S\subseteq V$ with $|S|=r$, the graph $G\setminus S^\perp$ is obtained by deleting $S^\perp$. The Gvozdenović–Laurent bound. For every $r\geq1$,

$$\vartheta^{(r)}(G)\leq r+\max_{S\subseteq V,\ S\text{ stable},\ |S|=r}\vartheta^{(0)}(G\setminus S^\perp).$$

The source states that this conjecture implies the main convergence conjecture. Its general validity is not established in the supplied text.