Quantitative max-sum matching conjecture for diametral balls

Let $d$ be a positive integer. Let $X$ be an even set of distinct points in $\mathbb R^d$ with minimum distance $r>0$, and let $\mathcal M$ be a max-sum matching of $X$. The closed balls induced by $\mathcal M$ are the balls having the matched pairs as diameters. Quantitative max-sum matching conjecture. There exists a constant $\varepsilon_d>0$ such that the intersection of the closed balls induced by $\mathcal M$ contains a ball of radius $\varepsilon_d r$. This is a quantitative strengthening of the question of whether every max-sum matching is an open Tverberg graph. The corresponding qualitative problem is known affirmatively in dimension $2$ but remains open in general dimensions, and the quantitative assertion is posed as a conjecture.