Conjecture on fractional chromatic number versus edges in triangle-free graphs

Let $G$ be a triangle-free graph with $m$ edges, and let $\chi_f(G)$ denote its fractional chromatic number. Edge-based fractional chromatic-number conjecture. As $m\to\infty$, every such graph satisfies

$$\chi_f(G)\leq (2^{4/3}+o(1))m^{1/3}/(\log m)^{2/3}.$$

The conjecture is presented as an analogue of Shearer's bound, giving a sharper fractional colouring prediction in terms of the number of edges.