Edwards–King fractional chromatic conjecture

For a graph $Q$ and $r\in\mathbb{N}\cup\{\infty\}$, let $\mathcal{C}_r(Q)$ consist of the maximal cliques of size less than $r$ together with the cliques of size exactly $r$, and define

$$\gamma_r(Q)=\max_{X\in\mathcal{C}_r(Q)}\frac{1}{|X|}\sum_{v\in X}\frac{d(v)+\omega(v)+1}{2}.$$

Write $\chi_f(Q)$ for the fractional chromatic number. Edwards–King fractional chromatic conjecture. If $Q$ is any graph, then

$$\chi_f(Q)\le\gamma_\infty(Q).$$

The source attributes this conjecture to Edwards and King; no resolution is given in the supplied text.