Intersecting ellipses induced by a max-sum matching

For an even set of points in the plane, choose a max-sum matching, that is, a perfect matching maximizing the sum of Euclidean distances of its edges. For each edge of the max-sum matching, consider the ellipse with foci at the edge's endpoints and eccentricity $2/\sqrt 3$. Using an optimization approach, we prove that the convex sets bounded by these ellipses intersect, answering a Tverberg-type question of Andy Fingerhut from 1995.