Zhai–Lin–Shu spectral Turán conjecture for books

Let $B_{r+1}$ denote the graph consisting of $r+1$ triangles sharing a common edge, and let $\rho(G)$ be the spectral radius of a graph $G$. A graph is $B_{r+1}$-free if it contains no copy of $B_{r+1}$. Zhai–Lin–Shu's conjecture. Let $G$ be a $B_{r+1}$-free graph on $m$ edges, where $mm_{0}(r)$ for sufficiently large $m_{0}(r)$. Then

$$\rho(G)\leq\sqrt{m},$$

with equality if and only if $G$ is a complete bipartite graph. The conjecture was confirmed by Nikiforov's theorem, so it is solved.