Dehmer–Pickl–Shi–Yu conjecture comparing degree-power and Laplacian spectral distances

Let $T$ and $T'$ be two trees on $n$ vertices. Define the degree-power index

$$F_2=\sum_{v\in V}\deg^2(v).$$

Let $d_{F_2}(T,T')$ and $d_{q_1}(T,T')$ denote the graph distance measures induced by $F_2$ and by the largest Laplacian eigenvalue $q_1$, respectively.

Dehmer–Pickl–Shi–Yu conjecture. For every such pair of trees,

$$d_{F_2}(T,T') \geq d_{q_1}(T,T').$$

The conjecture is refuted by the computational search reported in the paper: counterexamples occur already for $n=6$, and the number of counterexamples increases for the listed values of $n$.