Generalized-book spectral extremal conjecture

Let $B_{r,k}=K_r\vee I_k$ be the generalized book formed by joining every vertex of a clique $K_r$ to every vertex of an independent set $I_k$. Let $G$ be a graph with $m$ edges, and let $\lambda(G)$ be its adjacency spectral radius. Generalized-book conjecture. For $m$ sufficiently large, if $G$ is $B_{r,k}$-free, then

$$\lambda(G)\le \sqrt{\left(1-\frac{1}{r}\right)2m},$$

with equality if and only if $G$ is a complete bipartite graph for $r=2$, or a complete regular $r$-partite graph for $r\ge 3$, possibly together with isolated vertices. The conjecture generalizes the clique-free spectral bound and the book case; the source gives no resolution.