Planar support graph conjecture for shrinking rules C1, C2, and C3

Let $G$ be a graph, and let $(y,x)\in P^G_A$ be a vector. The support graph $\bar{G}$ of $(y,x)$ is the graph associated with this vector. Planar support graph conjecture. If $\bar{G}$ is planar, then the combination of the rules C1 and C2 dominates the rule C3. This conjecture is motivated by computational observations on planar support graphs, where rule C3 did not simplify the graph beyond the simplifications obtained using C1 and C2. The paper notes that the rules correspond to edge contractions, which preserve planarity, but does not establish the conjecture.