The bounded fractional-chromatic-number-to-Hall-ratio conjecture

Let $G$ be a graph. Its fractional chromatic number is

$$\chi_{\mathrm{f}}(G)=\sup_{w:V\to\mathbf R_{+}}\frac{w(V)}{\alpha(G,w)},$$

where $\alpha(G,w)$ is the maximum weight of an independent set and $w(V)=\sum_{v\in V}w(v)$. Its Hall ratio is

$$\rho(G)=\sup_{U\subseteq V}\frac{|U|}{\alpha(G[U])}.$$

The bounded fractional-chromatic-number-to-Hall-ratio conjecture. There is an absolute constant such that, for every graph $G$,

$$\chi_{\mathrm{f}}(G)\leq O(\rho(G)).$$

This is presented as the opposite of Johnson's conjecture, which concerned whether the ratio could be unbounded; it is open in the source.