Positivity conjecture for permanents of polystochastic matrices

A $d$-dimensional polystochastic matrix of order $n$ is a nonnegative array whose sums over every line are equal to $1$. Its permanent is

$$\operatorname{per} A=\sum_{\ell\in D(A)}\prod_{\alpha\in\ell}a_{\alpha},$$

where $D(A)$ is the set of diagonals of $A$, and a diagonal is a collection of $n$ indices such that no two share the same hyperplane. Positivity conjecture. Every polystochastic matrix of even dimension or odd order has a positive permanent. For odd dimension at least $3$ and even order, examples of polystochastic matrices with zero permanent are known, while no such matrices are known for the remaining parameter values; the conjecture asserts positivity in those remaining cases.