Ashraf's signless Laplacian eigenvalue-sum conjecture

Let $G$ be a graph with $n$ vertices and $e(G)$ edges, let $Q(G)$ be its signless Laplacian matrix, and let $S_k(Q(G))$ denote the sum of the $k$ largest eigenvalues of $Q(G)$. Ashraf's conjecture. For every $1\leq k\leq n$,

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

The conjecture is known for several classes, including graphs with at most ten vertices, regular graphs, trees, unicyclic graphs, bicyclic graphs, and tricyclic graphs when $k\ne3$, as well as for the specified endpoint values of $k$.