Dow–Gibson's permanent lower-bound conjecture for convex combinations of $(d-1)$-permutation matrices

Let $A$ be a $d$-dimensional matrix of order $n$, let $\Lambda_{d-1}(d,n)$ denote the set of $(d-1)$-permutation matrices, let $\operatorname{Hull}(\Lambda_{d-1}(d,n))$ be their convex hull, 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 conjecture. If

$$A\in\operatorname{Hull}(\Lambda_{d-1}(d,n)),$$

then

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

with equality if and only if

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

Dow and Gibson proved the conjecture for order $2$, but the general claim remains open.