Dehmer–Pickl–Shi–Yu conjecture on tree Laplacian and adjacency spectral distances

Let $T$ and $T'$ be two trees on $n$ vertices. Let $d_{q_1}(T,T')$ and $d_{\lambda_1}(T,T')$ denote the graph distance measures induced by the largest Laplacian eigenvalue $q_1$ and the largest adjacency eigenvalue $\lambda_1$, respectively.

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

$$d_{q_1}(T,T') \geq d_{\lambda_1}(T,T').$$

The paper states that this conjecture is disproved by families of counterexamples satisfying $d_{q_1}(T,T')=0<d_{\lambda_1}(T,T')$, so the proposed comparison fails even when the trees are Laplacian-cospectral but have different adjacency spectral radii.