The weak-similarity characterization of maximal centers of distances

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

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

and

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

where $C(X)$ denotes the center of distances of $(X,d)$. A bijection $\Phi:X\to Y$ is a weak similarity if there is a strictly increasing function $f:D(Y)\to D(X)$ such that

$$d(x,y)=f\bigl(\rho(\Phi(x),\Phi(y))\bigr)$$

for all $x,y\in X$.

The weak-similarity characterization. The equality

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

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

This conjecture asks whether, under the maximal-cardinality condition on the center of distances, equality of the cardinalities of the centers characterizes weak similarity. The supplied text gives no resolution status.