The random embedding edge-type conjecture for bridgeless cubic graphs

Let $G$ be a bridgeless cubic graph with $m$ edges, and let $(\pi,\lambda)$ be a random embedding of $G$. In such an embedding, an edge is bad singular or good singular according to the paper's facial-diagram classification, while a regular edge is an edge that is not singular.

Random embedding edge-type conjecture. The expected numbers of bad singular edges, good singular edges, and regular edges are respectively

$$\frac{m}{3},\qquad \frac{m}{3},\qquad \frac{m}{3}.$$

These expectations are proposed based on computer experiments and are intended to describe the distribution of edge types in random embeddings of bridgeless cubic graphs. The source provides experimental evidence but no resolution of the conjecture.