Oliveira et al.'s extremal conjecture for the signless Laplacian spectrum

Let $G$ be a connected simple graph on $n$ vertices, let $S_2(G)$ denote the sum of its two largest signless Laplacian eigenvalues, and define

$$f(G)=e(G)+3-S_2(G).$$

Let $K^+_{1,n-1}$ denote the star graph with an additional edge. Oliveira et al.'s conjecture. For any graph $G$ on $n\geq 9$ vertices,

$$f(G)\geq f(K^+_{1,n-1}).$$

Equality holds if and only if $G\cong K^+_{1,n-1}$. The paper states that this conjecture is proved in the present work, so it is solved.