Fingerhut's max-sum matching conjecture

Let $S$ be an even set of points in the plane, and let $\{a_ib_i:i=1,\dots,n\}$ be a max-sum matching of $S$, where $|S|=2n$. A point $o$ is required to satisfy, for every matched pair, the sum of its distances to the pair's endpoints. Fingerhut's conjecture. There exists a point $o$ in the plane such that

$$\|a_i-o\|+\|b_i-o\|\leq \frac{2}{\sqrt 3}\|a_i-b_i\|\qquad\text{for all }i=1,\dots,n.$$

The factor $2/\sqrt 3$ is optimal, and the conjecture was introduced by Andy Fingerhut in connection with minimum Steiner stars; it was previously known with weaker factors and is proved in this paper.