Diagonal-extremality conjecture for multidimensional extremal matrices

Let $A$ be a multidimensional $(0,1)$-matrix. Call $A$ diagonally extremal if replacing any zero entry by one produces a matrix containing a diagonal, where a diagonal is a full set of entries with one index in each direction.

Diagonal-extremality conjecture. Every multidimensional extremal matrix is diagonally extremal.

Every diagonally extremal matrix is extremal by definition, so the conjecture says that the two classes coincide. The paper proves that all of its constructions produce diagonally extremal matrices but does not establish the converse in general.