Heckman–Thomas fractional colorability conjecture for planar subcubic triangle-free graphs

Let $G$ be a subcubic triangle-free planar graph, meaning that every vertex of $G$ has degree at most $3$, $G$ contains no triangle, and $G$ is planar. A graph is fractionally $r$-colorable when its fractional chromatic number is at most $r$.

Heckman–Thomas planar conjecture. Every subcubic triangle-free planar graph is fractionally $8/3$-colorable.

This is the planar analogue of the stronger fractional-colorability conjecture for all subcubic triangle-free graphs. The supplied material gives no resolution status for this planar conjecture.