Conjectural fractional-chromatic bound from a largest low-chromatic induced subgraph

Let $G$ be a graph, let $\chi(G)$ and $\omega(G)$ denote its chromatic and clique numbers, let $\chi_f(G)$ denote its fractional chromatic number, and let $H$ be a largest induced subgraph of $G$ satisfying $\chi(H)\leq\omega(G)$. Fractional-chromatic bound conjecture. There exists a constant $c>0$ such that, whenever $\chi(G)>\omega(G)$,

$$\chi_f(G)<c\left(\frac{\omega(G)|V(G)|}{|V(H)|}\right).$$

This proposed bound combines the paper's intuition about the triangle estimate with the preceding odd-wheel conjecture; no general proof or disproof is given.