The serpent nest conjecture for accordion complexes

Consider 2n points on the unit circle and a reference dissection D of the convex hull of the odd points. The accordion complex of D is the simplicial complex of subsets of pairwise noncrossing diagonals with even endpoints that cross a connected set of diagonals of the dissection D. In particular, this complex is an associahedron when D is a triangulation, and a Stokes complex when D is a quadrangulation. We exhibit a bijection between the facets of the accordion complex of D and some dual objects called the serpent nests of D, settling in particular a conjecture stated by F.~Chapoton (2016) in the case of Stokes complexes.