Dunbar et al.'s bondage-number conjecture for planar graphs

Let $G$ be a planar graph, and let $b(G)$ denote its bondage number and $\Delta(G)$ its maximum degree.

Dunbar et al.'s conjecture. If $G$ is a planar graph, then

$$b(G)\leq \Delta(G)+1.$$

This conjecture concerns the relationship between the bondage number and maximum degree in planar graphs. The surrounding text notes that the bound for planar chordal graphs may help address it; its resolution is not given here.