The neighborhood-tour conjecture for ASEP vertices

Let $P_{\text{ASEP}}^{n}$ be the asymmetric subtour elimination polytope, let $\boldsymbol{x}$ be a vertex of this polytope, and let $\mathcal{N}(\boldsymbol{x})$ denote its neighborhood in the vertex structure considered by the paper. A tour is a Hamiltonian cycle of the complete directed graph on the node set. Neighborhood-tour conjecture. For every vertex $\boldsymbol{x}$, the neighborhood $\mathcal{N}(\boldsymbol{x})$ contains at least one tour. This is presented as a conjecture based on preliminary computational tests; the supplied text does not establish it or indicate a resolution.