Ruskey's Hamiltonian-path conjecture for the graph of linear extensions

Let $P$ be a finite sign-balanced poset, and let $\mathcal G(P)$ be the graph whose vertices are the linear extensions of $P$ and whose edges correspond to adjacent switches. Ruskey’s conjecture. The graph $\mathcal G(P)$ has a Hamiltonian path. This conjecture concerns Gray-code generation of linear extensions; the source cites algorithmic work and surveys but does not report a resolution.