Conjecture on fractional domatic number of planar graphs by girth

For an integer $g\ge 3$, let $\mathcal{P}_g$ be the class of planar graphs $G$ with minimum degree $\delta(G)=2$ and girth at least $g$. Write

$$\operatorname{FDOM}(\mathcal{P}_g)=\inf\{\operatorname{FDOM}(G):G\in\mathcal{P}_g\}.$$

Planar girth fractional-domatic conjecture. For every integer $k\ge 1$, if $3k-1\le g\le 3k+1$, then

$$\operatorname{FDOM}(\mathcal{P}_g)=\frac{3k+1}{k+1}.$$

The value is known to be $2$ for $g\le 4$, $7/3$ for $5\le g\le 7$, and $5/2$ for $8\le g\le 10$. The conjecture predicts the general extremal value, suggested by cycles, and remains open beyond these established ranges.