The isometry characterization by centers of distances

Let $n\in\mathbb{N}$ and let $(X,d)$ and $(Y,\rho)$ be ultrametric spaces satisfying

$$|X|=|Y|=2^n$$

and

$$|C(X)|=n+1,$$

where $C(X)$ denotes the center of distances of $(X,d)$. Two ultrametric spaces are isometric if there is a bijection between them preserving all distances.

The isometry characterization. The equality

$$C(X)=C(Y)$$

holds if and only if $(X,d)$ and $(Y,\rho)$ are isometric.

This conjecture proposes that, under the stated cardinality and maximal-center hypotheses, the center of distances determines the ultrametric space up to isometry. The supplied text gives no resolution status.