Positivity of permanents for even-dimensional or odd-order polystochastic matrices

Prove that every polystochastic matrix of even dimension or odd order has a positive permanent.

Background

The permanent-positivity problem is motivated by the fact that, although zero-permanent multidimensional permutations exist for odd dimension and even order, no zero-permanent polystochastic matrices are known for the complementary parameter regimes. The author formulates the unresolved assertion as a conjecture: every polystochastic matrix has positive permanent whenever its dimension is even or its order is odd. The paper further observes that it would suffice to establish positivity for all vertices of the corresponding Birkhoff polytopes.