Let $A$ and $B$ be two $n \times n$ real matrices such that:

- $\forall i, j: a_{ij} \geq 0, b_{ij} \geq 0$
- let $a_\max$ be the largest entry of $A$ and $b_\min$ be the smallest nonzero entry of $B$; for some positive $K$, $b_\min \geq K \cdot a_\max$

I am interested in a statement of the type

For $A$ and $B$ as above and for $K$ large enough, there exists a $k \times k$ submatrix $C$ of $A+B$ s.t. $|\det(C)|^{1/k} \geq |\det(A)|^{1/n}$

I know that by Minkowski's determinant theorem, if $A$ and $B$ are hermitian positive semidefinite, then $\det(A + B)^{1/n} \geq \det(A)^{1/n} + \det(B)^{1/n}$.

Do you know any other related facts, possible counterexamples, helpful intuitions?

Just for information, in the special case I am interested in, $A \in \{0, 1\}^{n \times n}$ and $B \in (2\mathbb{Z})^{n \times n}$.