The planar positive-valued cover conjecture

Let $G$ be a planar graph and let $(H,f)$ be a positive-valued cover. For each vertex $v\in V(G)$, suppose that $s\geq 2$ and

$$f(v,1)+f(v,2)+\dots+f(v,s)\geq 5.$$

Planar positive-valued cover conjecture. Under these assumptions, $H$ has a strictly $f$-degenerate transversal.

This conjecture would remove the restriction that the range of $f$ is contained in $\{0,1,2\}$ from the preceding results and would imply Thomassen's theorems that every planar graph can be partitioned into an independent set and a $3$-degenerate graph, and into a forest and a $2$-degenerate graph. Its resolution status is not specified in the supplied source context.