The Nordhaus–Gaddum full Brouwer conjecture for Laplacian spectra

Let $G$ be a graph with $n$ vertices, let $\overline{G}$ be its complement, and let $s_k(G)$ denote the sum of the $k$ largest Laplacian eigenvalues of $G$. Nordhaus–Gaddum full Brouwer conjecture. For every $k\in\{1,2,\cdots,n-1\}$,

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

with equality if and only if $G$ is a threshold graph having $n=2k+1$ vertices and clique number $k+1$. This is proposed as a Nordhaus–Gaddum analogue of the full Brouwer conjecture; the source presents only partial solutions, so the general assertion remains open.