Complementary small fibres of a shared repair force strictness

Write $d(x)=|x-\operatorname{round}(x)|$ for distance to the nearest integer. For finite sets $R,W\subseteq\mathbb N$, let $S(n,R,W)=\{v\in\mathbb N:0<v<n,\ v\notin R\}\cup W$. A strict time means a real $t$ such that $d(vt)>1/n$ for every $v\in S(n,R,W)$. Write $U_r=\{0\le b<r:\gcd(r,b)=1\}$, $F_{n,r}=\{n-r\le b<n:\gcd(r,b)=1\}$, and $B(n,r,m,q,b)$ for the existence of an integer $D$ with $|nmD-q|\le mr$ and $r\mid Db-q$. An active fibre for $(n,r,m,q)$ consists of natural numbers $(d,h,U,V)$ with $d,V>0$, $h\ge2$, $r=dh$, $q=dU$, $\gcd(h,U)=\gcd(h,V)=1$, $|nmV-U|\le hm$, and, for every $b\in U_r$, $B(n,r,m,q,b)$ if and only if $h\mid Vb-U$. Let $F,G$ be active fibres for $(n,r,m,q)$ and $(n,r,m,u)$, and let $n,r,s,m,k,q,u\in\mathbb N$. Assume $r\ge7$, $n\le2r$, $r<s<n$, $r\in R$, $m,k>0$, $mr=ks$, $mr<q<u$, and $F.h,G.h\in\{3,4,6\}$. If for every $c\in U_r$, $B(n,r,m,u,c)$ implies $\neg B(n,r,m,q,c)$, then $S(n,R,\{mr,q,u\})$ has a strict time.