I'm interested if the following is true:
Let $n> k\geq1$ be integers, let $A\in\mathbb Z^{k\times n}$ and denote the $\binom nk$ $k\times k$ minors of $A$ by $A_1,\ldots,A_N$. Then the equation $$\left|\begin{array}{c}A\\\hline X\end{array}\right|=\gcd(A_1,\ldots,A_N)$$ always has a solution $X\in\mathbb Z^{n-k\times n}$.
This is a generalisation of Can the determinant of an integer matrix with a given row be any multiple of the gcd of that row? (which is the case $k=1$). For $\style{text-decoration:line-through}{k=n}$ and $k=n-1$ the above conjecture is also (obviously) true.
Note that, as in the linked question, finding the minimal positive value of that determinant is equivalent to finding the minimal hypervolume of an $n$-dimensional simplex whose vertices are in an integer lattice and $k+1$ vertices are fixed.
Idea: The case $\gcd(A_1,\ldots,A_N)=1$ may follow from Elementary proof that if $A$ is a matrix map from $\mathbb{Z}^m$ to $\mathbb Z^n$, then the map is surjective iff the gcd of maximal minors is $1$