  Let $G$ be a simple graph with $n$ vertices and $e(G)$ edges, and $q_1(G)\geq
q_2(G)\geq\cdots\geq q_n(G)\geq0$ be the signless Laplacian eigenvalues of $G.$
Let $S_k^+(G)=\sum_{i=1}^{k}q_i(G),$ where $k=1, 2, \ldots, n.$ F. Ashraf et
al. conjectured that $S_k^+(G)\leq e(G)+\binom{k+1}{2}$ for $k=1, 2, \ldots,
n.$ In this paper, we give various upper bounds for $S_k^+(G),$ and prove that
this conjecture is true for the following cases: connected graph with
sufficiently large $k,$ unicyclic graphs and bicyclic graphs for all $k,$ and
tricyclic graphs when $k\neq 3.$
