Baker–Wang conjecture on Bernardi and rotor-routing torsors

Let $G$ be a connected ribbon graph without loops or multiple edges. For each vertex $v$ of $G$, let $b_v$ and $r_v$ denote respectively the Bernardi and rotor-routing torsors on the set of spanning trees of $G$ for its Picard group.

Baker–Wang conjecture. The torsors $b_v$ and $r_v$ agree for all vertices $v$ if and only if $G$ is planar.

Baker and Wang proved that the two torsors agree when $G$ is planar; this conjecture asserts the converse, namely that they disagree for every non-planar ribbon graph. The paper proves the conjecture.