A proof of the maximum Laplacian energy conjecture for connected graphs via a sharp eigenvalue-sum bound

Let $S_k(G)$ denote the sum of the $k$ largest Laplacian eigenvalues of a connected graph $G$ of order $n$ and size $m$. Write $\mathrm{PA}_{n,ω}$ for the graph obtained from an $ω$-vertex clique by attaching $n-ω$ pendant vertices to one of its vertices, and set \[ M_{n,k}:=\binom{k+1}{2}+n-k-1, \]