Given coprime $p,q=\Theta(\ell)$ and integer $t\geq3$ is there coprime $m,n=\Theta(\ell^{t-1})$ with $(mn,pq)=1$ such that there are $\alpha_i\in\Bbb Z$ and $m'\in\Bbb Z$ with $$\alpha_1p^{t-2}q^t+1=m'm^2$$ $$\alpha_2p^{t-1}q^{t-1}+1=m'mn$$ $$\alpha_3p^tq^{t-2}+1=m'n^2$$ where $m,n$ where $0<m'<p^{t}q^t$ holds?

Reduces to $$1\equiv m'm^2\bmod p^{t-2},\quad1\equiv m'mn\bmod p^{t-1},\quad1\equiv m'n^2\bmod p^{t}$$ $$1\equiv m'm^2\bmod q^{t},\quad1\equiv m'mn\bmod q^{t-1},\quad1\equiv m'n^2\bmod q^{t-2}$$

$\alpha_i=O(\ell^{2t})$ holds.