The planar graph positive-valued cover degeneracy conjecture

Let $G$ be a planar graph and let $(H,f)$ be a positive-valued cover, meaning that the cover function $f$ is positive-valued. Let $\kappa\geq 2$. A strictly $f$-degenerate transversal is a transversal of $H$ inducing, in every nonempty subgraph, a vertex whose degree is less than its assigned $f$-value. The planar graph positive-valued cover conjecture. If

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

for each $v\in V(G)$, then $H$ has a strictly $f$-degenerate transversal. This would extend the stated partition theorem and its corollary from two degenerate parts to arbitrary positive-valued covers with at least two parts; the conjecture remains open.