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This confirms a conjecture by Angenent-Daskalopoulos-Sesum. On the other hand, for every $k\\geq 2$ we also construct a $(k-1)$-parameter family of uniformly $(k+1)$-convex ancient ovals that are only $\\mathbb{Z}^{k}_{2}\\times \\mathrm{O}(n+1-k)$-symmetric. 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