Randić–degree-power entropy distance conjecture for trees

Let $T$ and $T'$ be any two trees with $n$ vertices. Randić–degree-power entropy distance conjecture. It holds

$$d_R(T,T')\geq d_{If_1}(T,T').$$

Here $R$ is the Randić index and $If_1$ is the degree-power entropy obtained from $If_k$ by setting $k=1$. The conjecture was verified for trees with a small number of vertices, but the paper gives counterexamples with more than $12$ vertices.