Lets say a linear form $ax+by$ represents $n$ if $ax+by=n$ for some $x,y≥0$.

Call a pair $a,b\in\Bbb N$ with $\mathsf{gcd}(a,b)=1$ a good coprime pair if linear form $ax+by$ represents both $rs$ and $tu$ for some $r,s,t,u>1$ implies it does not represent at least one of $rt$ and $su$ and at least one of $ru$ and $st$ where $rs,tu,ru,st,rt,su<g(a,b)$ ($g(a,b)$ is Frobenius number) holds.

Is there an $n_0\in\Bbb N$ such that for every $n\in\Bbb N_{>n_0}$ we have a good coprime pair in $[n,2n]$?

A bad pair example:

$a=22,b=21,s = 16, t = 17,r = 19,u = 15$

$$10a+4b=rs$$ $$8a+7b=rt$$ $$9a+2b=su$$ $$3a+9b=tu$$