Minimal size conjecture for $k$-positivity tests

Let a $k$-positivity test be a collection of matrix minors whose positivity guarantees that every minor of size at most $k$ is positive. For an $n\times n$ matrix, the size of a test is the number of minors in the collection. Minimal size conjecture. For any $k$, the minimal size of a $k$-positivity test is $n^2$. This extends the known cases $k=1$ and total positivity, both of which require $n^2$ tests; the conjecture asserts that no smaller test exists for any $k$.