Uniqueness conjecture for optimal hyperplane covers of extremal matrices

Let $A$ be a multidimensional $(0,1)$-matrix. An extremal matrix is one with no polydiagonal that becomes to contain a polydiagonal after changing any zero entry to one. An optimal hyperplane cover is a minimum-weight hyperplane cover of $A$.

Uniqueness conjecture. Every extremal matrix has the unique optimal hyperplane cover.

The conjecture asserts the proposed one-to-one correspondence between extremal multidimensional matrices and their optimal hyperplane covers. The paper reports no counterexamples and proves the claim for some classes of extremal matrices; its general status is not established here.