The six-forbidden-graphs fractional chromatic conjecture

Let $G$ be a triangle-free subcubic graph, and let $F_{11}$, $F_{14}^{(1)}$, $F_{14}^{(2)}$, $F_{19}^{(1)}$, $F_{19}^{(2)}$, and $F_{22}$ be the six specified graphs. Let $\chi_f(G)$ denote the fractional chromatic number of $G$. Fractional six-forbidden-graphs conjecture. If none of these six graphs occurs as a subgraph of $G$, then

$$\chi_f(G)\leq \frac{8}{3}.$$

This is the fractional analogue of the six-forbidden-graphs independent-set conjecture and would extend the $8/3$ bound beyond the planar setting. The source gives no resolution status for this conjecture.