On symmetric realizations of the simplicial complex of 3-crossing-free sets of diagonals of the octagon

Motivated by the question of the polytopal realizability of the simplicial complex Γn,k of (k + 1)-crossing-free sets of diagonals of the convex n-gon, we study the first non-obvious case, namely the simplicial complex Γ8,2 of 3-crossing-free sets of diagonals of the convex octagon. We give a complete description of the space of symmetric realizations of Γ8,2, that is, of the polytopes P whose boundary complex is isomorphic to Γ8,2, and such that the natural action of the dihedral group on Γ8,2 defines an action on P by isometry.