Facet-Hamiltonicity of type B permutahedra

A type $B$ permutahedron is the Coxeter permutahedron associated with the type $B$ reflection group; a polytope is facet-Hamiltonian when its skeleton contains a cycle that visits each facet in the required consecutive manner. Type $B$ permutahedron conjecture. Type $B$ permutahedra of all dimensions are facet-Hamiltonian. The authors report constructions in dimensions $3$ and $4$ and conjecture that the corresponding rhombic strips exist in every dimension; the general case remains open.