Sufficiency of the NR-facet condition for parsimonious relaxations

Let $P_n$ be the Graphical Traveling Salesman Polyhedron, let $\mathcal G_B$ be the graph associated with a relaxation $\mathcal R_B$, and call a facet of $P_n$ an NR-facet when it has the stated non-rank property. Parsimonious-relaxation conjecture. If every connected component of $\mathcal G_B$ contains a vertex corresponding to an NR-facet of $P_n$, then the relaxation $\mathcal R_B$ has the parsimonious property. This conjectures that the necessary condition established in the cited theorem is also sufficient; the supplied text gives no resolution.