Grötschel–Padberg diameter conjecture for the symmetric traveling salesperson polytope

Let $HC_n$ be the set of Hamiltonian cycles on $n$ vertices, and let

$$\TSP(n)=\operatorname{conv}\{x^v\mid x\in HC_n\}$$

be the symmetric traveling salesperson polytope, where $x^v$ denotes the characteristic vector of the Hamiltonian cycle $x$. Its skeleton is the graph whose vertices are the vertices of $\TSP(n)$ and whose edges are its one-dimensional faces. Write $d(G)$ for the maximum edge distance between any pair of vertices of a graph $G$.

Grötschel–Padberg conjecture. For every integer $n\geq 5$, the diameter of the skeleton of the symmetric traveling salesperson polytope is

$$d(\TSP(n))=2.$$

This conjecture is motivated by the study of edge-following algorithms for linear programming and by the relationship between polytope diameter and the Hirsch conjecture. It is based on complete descriptions for $n\leq 9$ and remains open.