Distance-set rigidity conjecture for finite GH-spaces generated by labeled trees

Let $i=1,2$ and let $T_i=T_i(l_i)$ be finite labeled trees with non-degenerate labelings $l_i:V(T_i)\to\mathbb{R}^+$. Suppose that $(V(T_1),d_{l_1})$ and $(V(T_2),d_{l_2})$ are $\bf GH$-spaces. For a space $(V(T_i),d_{l_i})$, let $D(V(T_i))$ denote its distance set. Distance-set rigidity conjecture. The following statements are equivalent:

1. $D(V(T_1))=D(V(T_2))$.
2. $(V(T_1),d_{l_1})$ and $(V(T_2),d_{l_2})$ are isometric.

The implication from isometry to equality of distance sets is immediate, and the converse is known when both trees have at most three vertices; the general case remains open.