H-free extension of Brouwer's Laplacian eigenvalue-sum conjecture

Let $k\ge 1$, let $H$ be a graph, and let $\operatorname{ex}(n;H)$ denote the maximum number of edges in an $n$-vertex $H$-free graph. For a graph $G=(V,E)$, define

$$\operatorname{\varepsilon}_k(G)=\sum_{i=1}^k\lambda_i(L(G))-|E|.$$

H-free Brouwer conjecture. For every $H$-free graph $G=(V,E)$ with $|V|\ge k$,

$$\operatorname{\varepsilon}_k(G)\le \operatorname{ex}(k+1;H).$$

The paper notes that the proposed bound is sharp, using an $(k+1)$-vertex extremal $H$-free graph together with isolated vertices. The supplied status does not indicate whether the conjecture is resolved.