Wiener–Randić distance conjecture for trees

Let $T$ and $T'$ be any two trees with $n$ vertices. Wiener–Randić distance conjecture. It holds

$$d_W(T,T')\geq d_R(T,T').$$

The conjecture compares graph distance measures induced by the Wiener and Randić indices. It was verified for trees with a small number of vertices, but the paper gives counterexamples with more than $12$ vertices.