Tree equidistant dimension conjecture

Let $T$ be a tree of order $n$, and let $P_n$ denote the path on $n$ vertices. The equidistant dimension of a graph, denoted by $eqdim$, is the minimum cardinality of a distance-equalizer set.

Tree equidistant dimension conjecture.

$$eqdim(T)\leq eqdim(P_n).$$

The conjecture proposes that among all trees of order $n$, the path has maximum equidistant dimension. The exact value of the equidistant dimension of trees is not known, although the path is suggested by the fact that every pair of vertices in a path has at most one equidistant vertex.