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For every $s>1$ and integer $d>4s$, when the regression function is $s$-H\\\"older, the unknown design density is bounded above and away from zero, and the conditional error laws may depend on the design but have mean zero, a common variance, and uniformly bounded fourth moments, we show that the minimax root-mean-square risk is bounded below by $n^{-\\beta}$ with $\\beta=\\frac{d(3s+1)+8s}{(d+2s)(d+4)}$. 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