Jia–Song's second distance signless Laplacian conjecture

Let $G$ be a connected graph of order $n\geq 4$, with remoteness $\rho$, and suppose that $G\ncong K_n$ and $G\ncong K_n-e$. Here $K_n-e$ is the complete graph with one edge removed, $K_n-2e$ is the complete graph with two matching edges removed, and $\partial_2$ denotes the second distance eigenvalue. Jia–Song's conjecture.

$$\rho+\partial_2\geq \frac{n}{n-1}+\frac{n-1-\sqrt{(n-1)^2+8}}{2},$$

with equality if and only if $G\cong K_n-2e$, where $2e$ are two matching edges. This conjecture is presented after several related theorems of Jia and Song; the excerpt gives no evidence that it has been resolved.