Random list threshold conjecture for Latin hypercubes

For $d\in\mathbb N$, a $d$-dimensional order-$n$ Latin hypercube is a $d$-dimensional array indexed by $[n]^d$ with entries from $[n]$, such that every line contains each symbol exactly once. For each $\mathbf{x}\in[n]^d$, let $L_{\mathbf{x}}\subseteq[n]$ be chosen independently by including each symbol with probability $p$. Latin-hypercube threshold conjecture. If $p=\omega(\log n/n)$, then with probability tending to one there exists a $d$-dimensional order-$n$ Latin hypercube $\mathcal L$ satisfying $\mathcal L_{\mathbf{x}}\in L_{\mathbf{x}}$ for every $\mathbf{x}\in[n]^d$. The cases $d=1$ and $d=2$ connect this claim to perfect matchings and Latin squares, respectively, while the general case remains open in the source.