Newton-Okounkov bodies obtained from certain orbits of plabic graphs

We investigate the plabic graphs corresponding to the quadrilateral Postnikov arrangements used by J.Scott to equip the homogeneous coordinate rings of Grassmannians with a cluster structure. More precisely we describe their orbits under the natural action of the dihedral group and show that the associated Newton-Okounkov bodies are all unimodular equivalent to Gelfand-Tsetlin polytopes.