The multidimensional permanent upper-bound conjecture

Let $A$ be a $d$-dimensional $(0,1)$-matrix of order $n$. For each $i$, let $r_i$ be the number of ones in the $i$th hyperplane of $A$. Multidimensional permanent upper-bound conjecture. The permanent of $A$ satisfies

$$\operatorname{per} A \leq n!^{d-2} \prod_{i=1}^n \left\lceil \frac{r_i}{n^{d-2}}\right\rceil!^{\frac{1}{\left\lceil r_i/n^{d-2}\right\rceil}}.$$

The bound was tested on a number of matrices of small order and dimension, but the source reports no general proof; it is proposed because existing estimates through hyperplane sums are inadequate and the known two-dimensional argument has not been successfully generalized.