One-parameter flexible polyhedron with all dihedral angles varying

Determine whether there exists a one-parameter flexible polyhedron in Euclidean 3-space that is homeomorphic to a closed surface without boundary, has no self-intersections, and whose every dihedral angle varies during a flex.

Background

The paper positively resolves Sabitov’s question by constructing a sphere-homeomorphic, self-intersection-free flexible polyhedron in Euclidean 3-space for which all dihedral angles change during a flex. However, the constructed example admits a 4-parameter family of flexes rather than a single-parameter family.

Motivated by the distinction between one-parameter and multi-parameter flexibility (as formalized in the literature on p-parametric flexible polyhedra), the authors explicitly pose the open problem of whether such a phenomenon—simultaneous variation of all dihedral angles—can occur in a strictly one-parameter flexible polyhedron without self-intersections.