The realization conjecture for finite distance centers of ultrametric spaces

Let $A$ be a finite subset of $\mathbb{R}^+$ with $0\in A$. For a finite ultrametric space $(X,d)$, let $D(X)$ be its distance set and let $C(X)$ be its center of distances.

The realization conjecture. There exists a finite ultrametric space $(X,d)$ such that

$$D(X)=C(X)=A.$$

This conjecture asks whether every finite set of nonnegative real numbers containing zero can occur simultaneously as the distance set and center of distances of a finite ultrametric space. The source mentions motivation from results of Mateusz Kula but gives no resolution status.