The toroidal graph without configurations in E DP-4-degeneracy conjecture

Let $G$ be a toroidal graph with no subgraph isomorphic to any configuration in the family denoted by $E$. Let $(H,f)$ be a cover of $G$, where $f$ assigns to each vertex of $H$ a value in $\{0,1,2\}$. 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 toroidal graph without configurations in $E$ conjecture. If

$$f(v,1)+f(v,2)+\dots+f(v,\kappa)\geq 4$$

for each $v\in V(G)$, then $H$ has a strictly $f$-degenerate transversal. The known theorem for a more restricted family of forbidden configurations motivates this proposed extension; its truth is not established in the supplied text.