Uniqueness conjecture for extremal connected graphs maximizing the two largest signless Laplacian eigenvalues

Let $G$ be a connected graph with $n$ vertices, let $c(G)=e(G)-n(G)+\omega(G)$ be its cycle-space dimension, and let $S_2(G)$ denote the sum of the two largest signless Laplacian eigenvalues of $G$. For $1\leq c(G)\leq n-2$, write $G(n-2-c(G),c(G))\cong G(s,n-2-s)$. Extremal graph conjecture. Among all connected graphs with $n$ vertices and $1\leq c(G)\leq n-2$, $G(n-2-c(G),c(G))\cong G(s,n-2-s)$ is the unique graph with maximal value of $S_2(G)$. The conjecture proposes uniqueness of the extremal graph in the signless-Laplacian analogue of the corresponding Laplacian result; the source does not provide evidence that it has been resolved.