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

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 $1.5$)?