Restoring deletions outside a separated cut preserves absence of strict times

Write $d(x)=|x-\operatorname{round}(x)|$ for distance to the nearest integer. For finite $R,W\subseteq\mathbb N$, put $S(n,R,W)=\{v\in\mathbb N:0<v<n,\ v\notin R\}\cup W$. A strict time is a real $t$ with $d(vt)>1/n$ for every $v\in S(n,R,W)$. A separated cut $C\subseteq W$ satisfies $p+q<n\gcd(p,q)$ for every $p\in C$ and $q\in W\setminus C$. Let $n\ge5$, let every $r\in R$ satisfy $n\le2r$ and $r<n$, and let $W$ be positive. Suppose $D\subseteq R$, $C\subseteq W$ is a separated cut, and every $r\in D$ has a smallest positive multiple in $W$ that lies in $C$. Then

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