Maximum permanent conjecture for nonnegative matrices of bounded rank

Let $A=(a_{ij})$ be an $n\times n$ nonnegative matrix of rank at most $k$, with row sums $r_1,\ldots,r_n$ and column sums $c_1,\ldots,c_n$. For $k,n\in\mathbb{N}$ with $1\leq k\leq n$, write $n=rk+s$ with $0\leq s<k$, and set $\vec r=(\underbrace{r,\ldots,r}_{k-s\text{ times}},\underbrace{r+1,\ldots,r+1}_{s\text{ times}})$. When all row sums are positive, let $\overline A^r$ be obtained by dividing each row of $A$ by its row sum; when all column sums are positive, define $\overline A^c$ analogously. Let $J_{\vec r}$ be the associated block-diagonal composition matrix and let $\mathfrak{S}_n$ denote the permutation matrices. Nonnegative permanent conjecture. One should have

$$\operatorname{per}(A)\leq \min\left(\prod_{i=1}^n r_i,\prod_{i=1}^n c_i\right)\left(\frac{r!}{r^r}\right)^{k-s}\left(\frac{(r+1)!}{(r+1)^{r+1}}\right)^s.$$

Equality should hold exactly in the four cases stated in the source: a zero row or column; the row-normalized matrix is a permutation of $J_{\vec r}$ when the positive row-sum product is smaller; the analogous column-normalized condition when the positive column-sum product is smaller; or both normalized conditions when the two products are equal. This is the row-sum and column-sum form equivalent to the stochastic-matrix conjecture.