Conjecture on fractional chromatic number of triangle-free graphs

Let $G$ be a triangle-free graph on $n$ vertices, and let $\chi_f(G)$ denote its fractional chromatic number. Fractional chromatic-number conjecture. As $n\to\infty$, every such graph satisfies

$$\chi_f(G)\leq (\sqrt{2}+o(1))\sqrt{n/\log n}.$$

The paper notes that the corresponding proved bound has leading constant $2+o(1)$ and conjectures that this factor can be improved for fractional chromatic number.