The metric dimension of the supertoken graph of the complete graph

Let ${\cal F}_k(K_n)$ be the supertoken graph whose vertices represent configurations of $k$ indistinguishable tokens on the complete graph $K_n$, with adjacency given by moving one token along an edge of $K_n$. The metric dimension of a graph is the minimum cardinality of a resolving set, namely a set of vertices whose distance vectors distinguish all vertices.

Metric-dimension conjecture. The metric dimension of the supertoken graph ${\cal F}_k(K_n)$ is

$$\operatorname{dim}({\cal F}_k(K_n))=n-1.$$

The preceding proposition establishes the upper bound $\operatorname{dim}({\cal F}_k(K_n))\leq n-1$; the conjecture asserts that this bound is sharp. The supplied text does not provide a lower-bound proof or evidence of resolution.