Maximum permanent conjecture for stochastic matrices of bounded rank

Let $\mathcal{M}_n^+$ denote the set of $n\times n$ nonnegative real matrices. Let $R_n^{\overline{k}}$ and $L_n^{\overline{k}}$ be the sets of row-stochastic and column-stochastic $n\times n$ matrices, respectively, of rank at most $k$. For $k,n\in\mathbb{N}$ with $1\leq k\leq n$, write $n=rk+s$ with $r,s\in\mathbb{N}$ and $0\leq s<k$. Let $J_m$ be the $m\times m$ doubly stochastic matrix with every entry equal to $1/m$, let $J_{\vec r}=J_r\oplus\cdots\oplus J_r\oplus J_{r+1}\oplus\cdots\oplus J_{r+1}$ for $\vec r=(\underbrace{r,\ldots,r}_{k-s\text{ times}},\underbrace{r+1,\ldots,r+1}_{s\text{ times}})$, and let $\mathfrak{S}_n$ be the set of permutation matrices. Maximum permanent conjecture. If $A\in R_n^{\overline{k}}\cup L_n^{\overline{k}}$, then

$$\operatorname{per}(A)\leq \left(\frac{r!}{r^r}\right)^{k-s}\left(\frac{(r+1)!}{(r+1)^{r+1}}\right)^s.$$

Equality should hold if and only if $A=PJ_{\vec r}Q$ for some $P,Q\in\mathfrak{S}_n$; in particular, every maximizing matrix should be doubly stochastic of rank $k$. This is formulated as equivalent to the corresponding conjecture for nonnegative matrices with prescribed row and column sums. The conjecture concerns the extremal permanent among stochastic matrices subject to a rank bound.