Intermediate fractional colorability conjecture for subcubic triangle-free graphs

Let $G$ be a subcubic triangle-free graph, and let $F_{14}^{(1)}$, $F_{14}^{(2)}$, $F_{11}$, and $F_{22}$ be the four specified graphs from the paper. Suppose that none of these graphs occurs as a subgraph of $G$. A graph is fractionally $r$-colorable when its fractional chromatic number is at most $r$.

Intermediate fractional colorability conjecture. Every such graph $G$ is fractionally $19/7$-colorable.

This conjecture is intermediate between the paper's $11/4$ theorem under exclusion of $F_{14}^{(1)}$ and $F_{14}^{(2)}$ and the stronger $8/3$ conjecture that additionally excludes $F_{11}$, $F_{22}$, $F_{19}^{(1)}$, and $F_{19}^{(2)}$. The supplied material gives no resolution status.