Let's say that the linear form $ax+by$ represents $n$ if $ax+by=n$ for some positive integer $x$ and $y$.

Call a pair $(a,b)\in\Bbb N\times\Bbb N$ with $\mathsf{gcd}(a,b)=1$ good if, for any $r,s,u,v>1$ with each of $rs,uv,ru,sv,rv,su<(a-1)(b-1)$ (the Frobenius number of $(a,b)$), there is at most one set from among $\{rs,uv\}$, $\{ru,sv\}$ and $\{rv,su\}$ with both components representable by $ax+by$.

Do good pairs exist at all?

If they do, then is it true that for every sufficiently large integer $l$, there is a good pair $(a,b)$ with $a,b\in[l,2l]$?

A bad pair example:

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

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

Related Some divisibility constraints in Frobenius coin problem