Let $C$ be a real positive definite matrix. Let $C_{i,j}$ denote the matrix we obtain by deleting the $i$-th row and the $j$-th column from $C$. I'd like to know if the following determinant inequality holds:
$$|C_{i,j}|^2 \leq |C_{i,i}||C_{j,j}|$$
I've checked a few hundred random examples, and haven't found a counterexample -- hopefully, this result is somewhere out there, if true.