The full Brouwer's Laplacian spectrum conjecture

Let $G$ be a graph on $n$ vertices with $m$ edges and Laplacian eigenvalues $\mu_1\geq\mu_2\geq\cdots\geq\mu_n$, and define

$$s_k(G):=\sum_{i=1}^k\mu_i.$$

For integers $k\in\{1,2,\cdots,n-1\}$, let $G_{k,r,s}$ be the graph of order $n=k+r+s$ consisting of a clique $K_k$ and two independent sets $\overline{K_r}$ and $\overline{K_s}$, with every vertex of $K_k$ adjacent to every vertex of $\overline{K_r}$, and with

$$N(v_i)\subsetneq V(K_k),\qquad N(v_{i+1})\subseteq N(v_i)$$

for $i=1,2,\cdots,s$ and $i=1,2,\cdots,s-1$, respectively. The full Brouwer's conjecture. For every such $k$,

$$s_k(G)\leq m+\binom{k+1}{2},$$

with equality if and only if $G\cong G_{k,r,s}$ for some $r\geq1$ and $s\geq0$. This is equivalent to requiring that equality holds exactly for threshold graphs with $n$ vertices and clique number $k+1$. The conjecture has been confirmed for graphs with at most $9$ vertices and for several values of $k$, but remains open in general.