Maximum cardinality of the center of distances of a finite ultrametric space

Let $n\geq1$ be an integer, let $\log_2(n)$ denote the binary logarithm of $n$, and let $\lfloor\log_2(n)\rfloor$ be its integer part. For an ultrametric space $(X,d)$, let $C(X)$ denote its center of distances. The center-of-distances cardinality conjecture. For every ultrametric space $(X,d)$ with $|X|=n$,

$$|C(X)|\leq1+\lfloor\log_2(n)\rfloor.$$

Moreover, there exists an ultrametric space $(Y,\rho)$ with $|Y|=n$ such that

$$|C(Y)|=1+\lfloor\log_2(n)\rfloor.$$

This conjecture proposes a sharp logarithmic upper bound for the center of distances of finite ultrametric spaces; the source gives no resolution status.