Amaro et al.'s strengthened signless Laplacian eigenvalue-sum conjecture

Let $G$ be a graph with $n\geq5$ vertices and $3\leq k\leq n-2$, let $Q(G)$ be its signless Laplacian matrix, and let $S_k(Q(G))$ denote the sum of its $k$ largest eigenvalues. Define $H_{n,k}$ to be the $P_3$-join graph isomorphic to $P_3[(n-k-1)K_1,K_{k-1},K_2]$. Amaro et al.'s conjecture.

$$S_k(Q(G))\leq S_k(Q(H_{n,k}))<e(G)+\binom{k+1}{2},$$

with equality if and only if $G=H_{n,k}$. This is presented as a strengthening of Ashraf et al.'s conjecture; the source records that the latter is proved in several graph classes, but does not give a resolution of this stronger claim.