The serpent nest conjecture for hollow quadrangulations

Let $\mathrm{Q}_\circ$ be a hollow quadrangulation. A serpent nest of $\mathrm{Q}_\circ$ is a potentially empty set of pairwise compatible serpents, and a maximal $\mathrm{Q}_\circ$-accordion dissection is a dissection maximal among those whose diagonals are compatible with the reference quadrangulation $\mathrm{Q}_\circ$.

Serpent nest conjecture. For any hollow quadrangulation $\mathrm{Q}_\circ$, there is a bijection between the serpent nests of $\mathrm{Q}_\circ$ and the maximal $\mathrm{Q}_\circ$-accordion dissections.

This conjecture generalizes the bijection between nonnesting partitions and triangulations of convex polygons from associahedral combinatorics to Stokes, or accordion, complexes. It was initially stated by F. Chapoton for reference hollow quadrangulations and is presented here as the specialization at $x=y=1$ of his more general conjecture.