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Yu et al. in [arXiv:2404.03423] conjectured that for $k\\geq2$ and $m$ sufficiently large, if $G$ is an $F_{2k+1}$-free or $F_{2k+2}$-free graph, then $\\lambda(G)\\leq \\frac{k-1+\\sqrt{4m-k^2+1}}{2}$ and the equality holds if and only if $G\\cong K_k\\vee\\left(\\frac{m}{k}-\\frac{k-1}{2}\\right)K_1$. Recently, Li et al. in [arXiv:2409.15918] showed that the above conjecture holds for $k\\geq 3$. The only left case is for $k=2$, which corresponds to $F_5$ or $F_6$. 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