Liu–Shu–Xue conjecture on distance spectral radius and maximum transmission

Let $G$ be a connected non-transmission-regular graph with $n$ vertices. Its distance spectral radius $\lambda_1(G)$ is the largest eigenvalue of its distance matrix, and $D_{\max}(G)$ is the maximum transmission among its vertices. Liu–Shu–Xue's conjecture.

$$D_{\max}(G)-\lambda_1(G)>\frac{1}{n+1}.$$

This conjecture gives a strict lower bound for the gap between maximum transmission and distance spectral radius in connected non-transmission-regular graphs. The supplied paper is a proof of the conjecture, so the claim is solved.