Ashraf et al.'s signless Laplacian eigenvalue-sum conjecture

Let $G$ be a connected simple graph on $n$ vertices, and let $S_k(G)$ denote the sum of the first $k$ largest signless Laplacian eigenvalues of $G$. Ashraf et al.'s conjecture. For any graph $G$ with $n$ vertices and any $k\in\{1,2,\ldots,n\}$,

$$S_k(G)\leq e(G)+\binom{k+1}{2}.$$

The conjecture is known for several classes, including regular graphs and connected triangle-free graphs when $k=2$, but remains open for general graphs when $k=2$.