A multidimensional nonnegative matrix is called polystochastic if the sum of entries in each line is equal to $1$. The set of all polystochastic matrices of order $n$ and dimension $d$ is a convex polytope $Ω_n^d$. In the present paper, we compare known bounds on the number of vertices of the polytope $Ω_n^d$ and prove that the number of vertices of $Ω_3^d$ is doubly exponential on $d$.