Existence of explicit counterexamples to 1-step refolding

Ascertain whether there exist explicit pairs of polyhedral manifolds of equal surface area that cannot be related by a single refolding step, thereby providing concrete counterexamples to universal 1-step refolding.

Background

Although the paper proves a universal 2-step refolding, it leaves open whether 1-step refolding can fail for specific pairs, beyond conjectural evidence.

Identifying such explicit counterexamples would establish the necessity of at least two refolding steps in concrete terms.