Baker and Wang's conjecture on Bernardi and rotor-routing actions

Let $G$ be a nonplanar ribbon graph with no multiple edges and no loops. The graph has a cyclic ordering of the edges incident to each vertex, and $\rho_q$ and $\beta_q$ denote respectively the rotor-routing and Bernardi actions of the degree-zero Picard group on the set of spanning trees of $G$.

Baker and Wang's conjecture. There exists a vertex $q$ of $G$ such that

$$\rho_q\neq\beta_q.$$

For planar ribbon graphs, the two actions define the same torsor structure and are independent of the base vertex. This conjecture asserts that for every nonplanar ribbon graph without multiple edges or loops, the two actions differ for at least one choice of base vertex.