Signless Brouwer conjecture for graphs

Let $G$ be a graph with $n$ vertices, and let $S_k(G)$ denote the sum of the $k$ largest signless Laplacian eigenvalues of $G$:

$$S_k(G)=\sum_{i=n+1-k}^{n}\lambda_i(Q(G)).$$

Signless Brouwer conjecture. For an integer $k$ with $1\leq k\leq n$,

$$S_k(G)\leq |E|+{k+1\choose 2}.$$

This conjecture is the signless-Laplacian analogue of Brouwer's conjecture for the sum of the largest Laplacian eigenvalues, and has been studied by many researchers. The source discusses it in the context of threshold graphs, but the stated conjecture concerns arbitrary graphs.