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?

There seems to be room for such $m',m,n,\alpha_i$. Such an $m'$ has to be of form $p^2q^2m''$ where $0<m''<p^{t-2}q^{t-2}$ holds.

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

The expected intersection of choices is $$\frac{\mbox{# choices of }\alpha_i\times\mbox{# choices of }m',m\mbox{ and }n}{(\mbox{# of choices of resiudes modulo }p^{t-2}q^{t-2})^3}\approx\frac{\ell^{6t}\times \ell^{2t-4}\ell^{t-1}\ell^{t-1}}{\ell^{6t-6}}=\ell^{4t}\gg1.$$