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 $$\alpha_1p^tq^t+1=m'm^2$$ $$\alpha_2p^tq^t+1=m'mn$$ $$\alpha_3p^tq^t+1=m'n^2$$ where $m,n$ where $0<m'<\ell^{2t}$ holds?