Polydiagonal-to-diagonal conjecture for hyperplane-cover matrices

Let $A$ be a multidimensional $(0,1)$-matrix, let $A(\Lambda)$ be the matrix defined by a hyperplane cover $\Lambda$, and suppose that $A$ contains a polydiagonal.

Polydiagonal-to-diagonal conjecture. Then $A$ contains a diagonal.

The source proposes this as a possible additional conjecture that would make the diagonal-extremality conjecture equivalent to its weaker minor version. Its general status is left open.