The densely sparse cluster characterization for bridgeless cubic graphs

Let $G$ be a bridgeless cubic graph. Write $K_G$ for its maximal conflicting subgraph, $r(G)$ for its resistance, and call $G$ a densely sparse cluster when it has the corresponding cluster structure defined in the paper. Densely sparse cluster conjecture.

$$K_G = G \quad\Longleftrightarrow\quad r(G)=2 \text{ and } G \text{ is a densely sparse cluster}.$$

The conjecture attempts to characterize bridgeless cubic graphs whose maximal conflicting subgraph is the whole graph. The supplied text does not establish a resolution of this claim.