A noncoprime active blocker has a reduced congruence fibre

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 $h\mid Vb-U$. For natural $n,r,m,q$, assume $0<r<n$, $m>0$, $mr<q$, $r\nmid q$, $\gcd(r,q)\ne1$, and $B(n,r,m,q,b)$ for some $b\in U_r$. Then an active fibre for $(n,r,m,q)$ exists.