Schalekamp–Williamson–van Zuylen conjecture on the subtour-LP integrality gap

Consider the metric traveling salesperson problem and its subtour linear programming relaxation. A feasible solution is half-integral if every edge variable satisfies $x_e\in\{0,1/2,1\}$. The integrality gap is the supremum, over instances, of the ratio between the cost of an optimal tour and the cost of an optimal subtour-LP solution. Schalekamp–Williamson–van Zuylen conjecture. The integrality gap for the subtour LP is attained on half-integral vertices of the polytope. Equivalently, half-integral LP solutions have the largest integrality gap among all fractional solutions. This conjecture identifies half-integral instances as the hardest instances for metric TSP. The paper's $1.49993$-approximation algorithm for half-integral solutions would imply the same upper bound for the subtour-LP integrality gap if the conjecture holds.