Given coprime $p,q$ and integer $t\geq3$. Consider the diophantine equations $$m'm^2=1+k_1p^{t-2}q^{t}$$ $$m'n^2=1+k_2p^tq^{t-2}$$ $$m'mn=1+k_3p^{t-1}q^{t-1}$$

Is there a solution where $m,n$ are coprime and $(mn,pq)=1$ and we get $m,n=O(\max({p^{t-1}},{q^{t-1}}))$?

From counting arguments it looks like there is room for $m,n=O(\max({p^{2(t-1)}},{q^{2(t-1)}}))$?