Finite-step refolding with convex intermediates for convex polyhedra

Determine whether, when both the source and target are convex polyhedra of equal surface area, there exists a finite-step refolding in which every intermediate polyhedron remains convex.

Background

The authors’ 2-step refolding result relies on potentially nonconvex intermediate manifolds. Maintaining convexity throughout the refolding is a significant additional constraint.

This question echoes a long-standing problem recorded in prior literature and remains unresolved in the context of multi-step refolding.