  We design a $1.49993$-approximation algorithm for the metric traveling
salesperson problem (TSP) for instances in which an optimal solution to the
subtour linear programming relaxation is half-integral. These instances
received significant attention over the last decade due to a conjecture of
Schalekamp, Williamson and van Zuylen stating that half-integral LP solutions
have the largest integrality gap over all fractional solutions. So, if the
conjecture of Schalekamp et al. holds true, our result shows that the
integrality gap of the subtour polytope is bounded away from $3/2$.
