Dow–Gibson's modified van der Waerden conjecture for 1-polystochastic matrices

Let $\Omega_1(d,n)$ denote the set of $1$-polystochastic $d$-dimensional matrices of order $n$, let $\operatorname{per_1}(A)$ denote the permanent, and let $J_n^d$ be the $d$-dimensional matrix of order $n$ with all entries equal to $1$. Dow–Gibson's modified conjecture. If $A\in\Omega_1(d,n)$, where $d$ is even or $n$ is odd, then

$$\operatorname{per_1}(A)\geqslant(n!)^{d-1}/n^n,$$

with equality if and only if

$$A=n^{-1}J_n^d.$$

The conjecture is known to fail for odd dimensions, asymptotically, and for $n=3$ with $d\geqslant3$; it therefore remains open only in the parameter ranges not excluded by these counterexamples.