Bounded path covering with forests number for non-crossing shortest paths

Let $G$ be a plane graph, and let $P$ be a set of non-crossing single-touch shortest paths in $G$. The path covering with forests number of $P$, denoted by $\operatorname{PCFN}(P)$, is the minimum number of forests needed to cover the paths in $P$. Bounded path covering conjecture. There exists $\ell\in\mathbb{N}$ such that

$$\operatorname{PCFN}(P)\leq\ell$$

for every set $P$ of non-crossing single-touch shortest paths in a plane graph. This asks whether the path covering with forests number is bounded by a universal constant when the paths' extremal vertices are not required to lie on the same face; the paper presents this as a generalization for future work and gives a related remark without proof.