On the sum of Laplacian eigenvalues of a signed graph

For a signed graph Γ, let e(Γ) denote the number of edges and Sk(Γ) denote the sum of the k largest eigenvalues of the Laplacian matrix of Γ. We conjecture that for any signed graph Γ with n vertices, Sk(Γ)≤e(Γ)+(k+12)+1 holds for k=1,…,n. We prove the conjecture for any signed graph when k=2, and prove that this conjecture is true for unicyclic and bicyclic signed graphs.