Martinsson–Steiner conjecture on the fractional chromatic number of degenerate triangle-free graphs

Let $d$ be sufficiently large, and let $G$ be a $d$-degenerate triangle-free graph with fractional chromatic number $\chi_f(G)$. Martinsson–Steiner conjecture. The following two assertions hold:

1. For every such graph $G$,

$$\chi_f(G)\leq(1+o(1))\frac{d}{\log d}.$$

2. There exists a triangle-free $d$-degenerate graph $G$ such that

$$\chi_f(G)\geq(1-o(1))\frac{d}{\log d}.$$

The upper-bound part sharpens the proved bound of Martinsson and Steiner, while the lower-bound part asserts its asymptotic sharpness. The source gives no resolution of this two-part conjecture.