Existence of a tropical Fermat–Weber point in tropical principal component analysis

Let $\mathcal U_N$ denote the space of ultrametrics, and let $D^{(1)},\dots,D^{(n)}$ be a sample of ultrametric trees. For $s\geq 1$, let the $s$th-order tropical principal component analysis (tropical PCA) of the dataset be the corresponding tropical principal polytope. A tropical Fermat–Weber point is a point minimizing the total tropical distance to the sample.

Existence conjecture. There exists a tropical Fermat–Weber point

$$x^*\in\mathcal U_N$$

that is contained in the $s$th-order tropical PCA of the dataset for every $s\geq 1$.

This question concerns whether a central representative of an ultrametric-tree dataset is always captured by tropical PCA at every order. The supplied text presents it as an open problem; no resolution is given.