The Maximum Diagonal Product of a Stochastic Matrix of Bounded Rank

Let A = (aij) be a nonnegative row-stochastic or column-stochastic n × n matrix with rank A ≤ k, where 1 ≤ k ≤ n. Write n = qk + s, where 0 ≤ s < k. The author conjectured in 2018 that, for every permutation σ of {1, …, n}, ∏i=1n ai,σ(i) ≤ q−q(k−s)(q+1)−(q+1)s. We prove this bound and classify all equality cases.