Random higher-dimensional permutation universality conjecture

For $d\geqslant 2$, a $d$-permutation of order $n$ is a $(d+1)$-array over $\{0,1\}$ of order $n$ in which every line contains a unique $1$. A $d$-pattern of order $k$ is a sequence $(\sigma_1,\dotsc,\sigma_d)$ with each $\sigma_\ell\in S_k$; a $d$-permutation is $k$-pattern-universal if it contains every such $d$-pattern. Random higher-dimensional permutation universality conjecture. For $d\geqslant 2$, there exists a constant $C>0$ such that a random $d$-permutation of order $Ck^{(d+1)/d}$ is $k$-pattern-universal with high probability as $k\to\infty$. Monotone-subsequence results imply the lower bound $\Omega(k^{(d+1)/d})$ for the order needed with high probability, so the conjecture predicts that this lower-bound scale is tight.