Is there an $n_0\in\Bbb N$ such that for every $n\in\Bbb N_{>n_0}$ there are $a,b\in\Bbb N$ with $n<a,b<2n$ with $\mathsf{gcd}(a,b)=1$ such that if $ax+by=rs$ and $ax'+by'=tu$ for some $x,y,x',y'>0$ then there is no $v,w,v',w'$ such that $av+bw=ru$ and $av'+bw'=st$?