The asymptotic enumeration conjecture for high-dimensional permutations

For positive integers $d$ and $n$, let $S_{d,n}$ denote the set of $d$-dimensional permutations of order $n$: equivalently, the $[n]^d$ arrays with entries in $[n]$ in which every line contains each element of $[n]$ exactly once. Thus $|S_{d,n}|$ is the number of $d$-dimensional permutations of order $n$.

Asymptotic enumeration conjecture.

$$|S_{d,n}|=\left((1+o(1))\frac{n}{e^d}\right)^{n^d}.$$

For $d=1$ this follows from Stirling's formula, and for $d=2$ it is the known asymptotic estimate for Latin squares. The paper's main result gives the corresponding upper bound, while the complementary lower bound remains open; the displayed equality is therefore an open conjecture.