Given $m,n\in\Bbb N$ and $\beta\in(0,1)$ consider the uniformly picked random matrix $A\in\Bbb Z^{n\times (n+1)}$ such that each entry of $A$ is in $\{0\}\cup\big([\beta m,m]\cap\Bbb N\big)$ and exactly one column 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?

Do we have enough cancellations so that the expected maximum absolute value is $cm^{\frac k2+\epsilon}$ for some $c>0$?