The principal conjecture on clique coverings and stable sets in B-graphs

Let $G$ be a B-graph, meaning that $\sigma_v(G)=\alpha(G)$, and suppose that $G$ has no isolated vertices. Let $\omega_e(G)$ be the greatest natural number such that every edge of $G$ belongs to a clique of size at least $\omega_e(G)$, and let $\sigma_v(G)$ denote the corresponding vertex-stable-set invariant. For every maximum stable set $M$,

$$\omega_e(G)\sigma_v(G)\leq |V|.$$

Moreover, there exist disjoint sets $A_j\subset V$, for $j=1,\ldots,|M|$, such that $|M\cap A_j|=1$ for every $j$, and $G[A_j]$ is a clique of order $\omega_e(G)$ for every $j$. This conjecture is part of the study of bounds relating clique-covering and stable-set invariants; weaker versions are known, but the full assertion is not resolved in the supplied text.