Klivans's uniqueness conjecture for sandpile torsor structures on plane graphs

Let $(G,\chi)$ be a plane graph, meaning a planar graph equipped with a plane ribbon structure. A sandpile torsor structure assigns a free transitive action of the sandpile group $\operatorname{Pic}^0(G)$ to the spanning trees $\mathcal T(G)$, independently of a choice of sink vertex. Klivans's conjecture. For plane graphs, there is only one sandpile torsor structure. The conjecture is motivated by the equivalence of the Bernardi and rotor-routing algorithms on plane graphs; in this paper, the authors make the notion precise through consistency and prove that there are exactly four consistent algorithms, all having the same structure as rotor-routing, thereby resolving the conjecture.