Assume $p,q=\Theta(\ell)$.

Given coprime $p,q$ and integer $t\geq3$. Consider the diophantine equations in variables $m',m,n,k_1,k_2,k_3$ $$m'm^2=1+k_1p^{t-2}q^{t}$$ $$m'n^2=1+k_2p^tq^{t-2}$$ $$m'mn=1+k_3p^{t-1}q^{t-1}$$ where $m,n$ are coprime with $(mn,pq)=1$ and $m,n=\Theta(\ell^{(t-1)})$.

Essentially given coprime $p,q$ and integer $t\geq3$ is there coprime $m,n$ with $(mn,pq)=1$ and $m,n=\Theta(\ell^{(t-1)})$ such that there is a common integer $m'$ serving as modular inverse of $m^2\bmod p^{t-2}q^t$, $n^2\bmod q^{t-2}p^t$ and $mn\bmod p^{t-1}q^{t-1}$?

We can get $m,n=\Theta(\ell^{2(t-1)})$. Can $2$ be replaced by $1$ (or any fixed real less than say $1.5$)?