Harris's conjecture on the Hall ratio and fractional chromatic number

Let $G$ be a graph, let $\chi_f(G)$ denote its fractional chromatic number, and let $\rho(G)$ denote its Hall ratio, the maximum of $v(H)/\alpha(H)$ over all non-null subgraphs $H\subseteq G$. Harris's Hall-ratio conjecture. There exists a constant $C$ such that

$$\chi_f(G)\le C\cdot\rho(G)$$

for every graph $G$. The conjecture proposed a universal constant-factor upper bound on the fractional chromatic number in terms of the Hall ratio, but the paper's constructions refute it by producing graphs for which the ratio $\chi_f(G)/\rho(G)$ is unbounded.