Facet-Hamiltonian conjecture for B-permutahedra

Let $P$ be a polytope of dimension $d$, and call each $(d-1)$-dimensional face a facet. A facet-Hamiltonian cycle in $P$ is a cycle $C$ in the skeleton of $P$ such that, for every facet $F$ of $P$, the intersection $C\cap F$ is connected and nonempty. The $B$-permutahedron has as vertices all signed permutations of $[n]$, and its edges join signed permutations differing either by an adjacent transposition preserving all signs or by complementing the sign of the first entry.

Facet-Hamiltonian conjecture. The $B$-permutahedron of any dimension $d\geq 2$ has a facet-Hamiltonian cycle.

The cited authors prove facet-Hamiltonian cycles for several families, including permutahedra and various associahedra, and paths for some graph associahedra. The assertion for every $B$-permutahedron of dimension at least $2$ remains open.