The planar and toroidal graphs without intersecting triangles DP-4-degeneracy conjecture

Let $G$ be a planar or toroidal graph without intersecting triangles. 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 planar and toroidal graphs without intersecting triangles 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. This conjecture extends established DP-colouring, list-colouring, and vertex-arboricity results for planar and toroidal graphs without intersecting triangles; the full cover statement remains open.