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$$ As an application, we apply the above to $M=\\mathbb{E}_{X \\sim \\mu}(\\langle X,\\theta\\rangle X\\otimes X)$ for $\\theta\\in S^{n-1}$ and show that the Kannan--Lov\\'asz--Simonovits constant $\\psi_n$ satisfies $$ \\psi_n\\leq C\\log^{1/4}n $$ for some absolute constant 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