Alternating orientations minimize arborescences in Eulerian planar graphs

Let $G$ be an Eulerian planar graph. An orientation is alternatingly in- and outward oriented at each vertex when, in the cyclic order around every vertex, its incident edges alternate between incoming and outgoing; equivalently, each facial cycle is oriented. Planar alternating-orientation conjecture. The orientation minimizing the number of arborescences is the alternatingly in- and outward oriented orientation at each vertex, namely the orientation in which the cycle around each face is oriented. The conjecture is motivated by the relation between directed short cycles and arborescence numbers. In the planar dual formulation, it says that the standard orientation of the associated bipartite dual corresponds to a facet of minimal volume of the symmetric edge polytope.