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Keevash and Sudakov improved this result to graphs with average degree at least $\\frac{2}{5}n$. We strengthen these results by showing that the conjecture holds for graphs with minimum degree at least $\\frac{5}{14}n$ and for graphs with average degree at least $(\\frac{2}{5} - \\varepsilon)n$ for some absolute $\\varepsilon >0$. 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