7 problems

Order-two essential-weight conjecture. The values $k_{\alpha}$, for $\alpha\in\operatorname{supp}(K)$, are essential weights of an optimal hyperplane cover for some multidimensiona…

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

Minor conjecture. The submatrix $A_{\alpha}$ contains a polydiagonal.

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

Divisibility conjecture. Every entry $\lambda_{i,j}$ is an integer multiple of $\delta$.

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

Let $A$ be a $d$-dimensional $(0,1)$-matrix of order $n$. For each $i$, let $r_i$ be the number of ones in the $i$th hyperplane of $A$. Multidimensional permanent upper-bound conje…