Spectral Turán conjecture for non-bipartite 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 $B_{r+1}$. Let $S_{m,1}^{+}$ be the graph obtained from $K_{1,m-1}$ by adding an edge within its independent set. Non-bipartite book conjecture. If $G$ is a non-bipartite $B_{r+1}$-free graph with $m$ edges, then

$$\rho(G)\leq\rho\big(S_{m,1}^{+}\big),$$

with equality if and only if $G\cong S_{m,1}^{+}$. This conjecture concerns the spectral extremal graph among non-bipartite book-free graphs; the source proposes it without giving a resolution.