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We prove that $R$ has bounded Chv\\'atal-Gomory rank (CG-rank) provided that $S$ has bounded notch and bounded gap, where the notch is the minimum integer $p$ such that all $p$-dimensional faces of the $0/1$-cube have a nonempty intersection with $S$, and the gap is a measure of the size of the facet coefficients of $\\mathsf{conv}(S)$.\n  Let $H[\\bar{S}]$ denote the subgraph of the $n$-cube induced by the vertices not in $S$. 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