Upper bounds on the permanent of multidimensional $(0,1)$-matrices

The permanent of a multidimensional matrix is the sum of products of entries over all diagonals.
By Minc's conjecture, there exists a reachable upper bound on the permanent of $2$-dimensional $(0,1)$-matrices. In this paper we obtain some generalizations of Minc's conjecture to the multidimensional case. For this purpose we prove and compare several bounds on the permanent of multidimensional $(0,1)$-matrices.
Most estimates can be used for matrices with nonnegative bounded entries.