The atom bound conjecture for inhibiting sets

Let a partial matrix have a pattern with variability $k$. An atom is a square submatrix whose determinant inequality is used in an inhibiting set, and an inhibiting set is a collection of such atoms whose simultaneous determinant inequalities have no solution although every proper subcollection has a solution.

Inhibiting-set atom bound conjecture. Every inhibiting set of a non-completable partial matrix with variability $k$ contains at most $k+1$ atoms.

The paper proves this bound when all atoms in the inhibiting set have convex positive solution sets, using Helly's theorem. Since atoms need not be convex, the general bound remains open.