The fractional chromatic bound for triangle-free degenerate graphs

Let $G$ be a triangle-free graph that is $d$-degenerate, meaning every subgraph of $G$ has a vertex of degree at most $d$. The fractional chromatic bound. Then

$$\chi_{\mathrm{f}}(G)\lesssim\frac{d}{\log d}.$$

The claim follows in the source from the proposed Hall-ratio conjecture together with a triangle-free coloring estimate; it is therefore presented as a conjectural consequence rather than proved independently there.