Given $m,n,\ell\in\Bbb N$ and $\beta\in(0,1)$ consider the uniformly picked random matrix $A\in\Bbb Z^{n\times (n+1)}$ with $0\neq|\mathsf{det}(A^\circ)|\leq m^{\frac 1\ell}$ where $A^\circ$ is the first $n\times n$ submatrix with entries in $\{0\}\cup\big([\beta m,m]\cap\Bbb N\big)$ and the last remaining column of $A$ having entries from $[\beta m^k,m^k]\cap\Bbb Z$ for some fixed $k>1$.

What is the distribution of the maximum absolute value of all possible $n\times n$ minor of such a matrix?

Can we expect enough cancellations so that the expected maximum absolute value is $cm^{\frac k\ell+\epsilon}$ for some $c>0$ at least when $\ell=2$?