Weeping Willow structure of restricted flip graphs

Prove the Weeping Willow Conjecture: every connected component of the n-vertex flip graph of triangulated 3-spheres that lies outside the polytopal closure descends from the polytopal closure of the m-vertex flip graph for some m greater than n through 4–1 vertex-removing flips, equivalently, show that every 1–4 flip applied to a triangulation in the polytopal closure of the n-vertex flip graph produces a triangulation in the polytopal closure of the (n+1)-vertex flip graph.

Background

The paper defines the polytopal closure as the connected component containing the boundary complexes of convex 4-polytopes. Its computational results show that the known isolated unflippable triangulations outside this closure enter the polytopal closure after one additional vertex is inserted.

The conjecture generalizes this observed behavior by proposing that every non-polytopal component is a downward branch from the polytopal closure at a larger vertex number. If true, the polytopal closure would form a central core of the restricted flip graph, with all other components arising through vertex-removing 4–1 flips.