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Extending a recent result of Norin and Yepremyan, we confirm this conjecture for graphs homomorphic to so-called Andr\\'asfai graphs. As a consequence, Erd\\H{o}s' conjecture holds for every triangle-free graph $G$ with minimum degree $\\delta (G)>10n/29$ and if $\\chi (G)\\leq 3$ the degree condition can be relaxed to $\\delta (G)> n/3$. 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