Optimality of two-step refolding (even for convex polyhedra)

Prove that two refolding steps are sometimes necessary even when both manifolds are convex polyhedra; equivalently, demonstrate the existence of a pair of convex polyhedra of equal surface area that do not admit any 1-step refolding, establishing the optimality of the 2-step upper bound.

Background

The main result of the paper shows that any two polyhedral manifolds of equal surface area are connected by a 2-step refolding, implying the refolding-graph has diameter at most two.

The authors conjecture that this bound is tight, even when restricted to convex polyhedra, and reference supporting evidence from conjectures about doubly covered triangles that suggest 1-step refoldings need not exist in general.