The firefighter number conjecture for planar graphs of girth at least 5

Let $\mathcal{P}_5$ be the class of planar graphs of girth at least $5$, and let $\mathit{ff}(\mathcal{P}_5)$ denote its firefighter number. Firefighter number conjecture for planar graphs of girth at least 5.

$$\mathit{ff}(\mathcal{P}_5)=1.$$

The paper explains that this would be stronger than the known result $\mathit{ff}(\mathcal{P}_9)=1$ and suggests adapting its proof techniques to the girth-$5$ case, but the conjecture remains open.