In an $m \times n$ matrix $X$, there are exactly $\binom{m}{r}\binom{n}{r}$ minors of size $r \times r$. Is it always possible to construct a real matrix $X$ such that the absolute value of the determinant of the $r \times r$ submatrices is arbitrarily ordered in modulus?
In other words, for $I \subset \{1,\dots,m\}$, $J \subset \{1, \dots, n\}$, let $X_{I,J}$ be the submatrix of $X$ with entries in positions $I \times J$. Given an ordering on the set $$\{I_i \times J_j \mid I_i \subset \{1,\dots,m\}, J_j \subset \{1, \dots, n\}, |I_i| = |J_j| = r\}$$ does there exist a matrix $X$ such that the set $\{|\text{det}(X_{I_i,J_j})|\}$ has the same ordering in modulus? Has this question been answered somewhere in literature that someone could point me towards?
