Energy–graph-entropy distance conjecture for trees

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

$$d_E(T,T')\geq d_{Ig}(T,T').$$

Here $E$ is the graph energy and $Ig$ is the entropy based on the absolute adjacency-matrix eigenvalues. The conjecture was verified for trees with a small number of vertices, but the paper gives counterexamples with more than $12$ vertices.