The threshold conjecture for Latin squares in random three-dimensional arrays

Let $M$ be sampled from the distribution $\mathcal{M}(n,n,n;p)$ of $n\times n\times n$ zero-one arrays whose entries are independently $1$ with probability $p$. A Latin square is a Latin box in this three-dimensional array, as defined in the paper. The threshold for $M\sim\mathcal{M}(n,n,n;p)$ to contain a Latin square is

$$p=\frac{\log n}{n}.$$

Latin-square threshold conjecture. The threshold for $M\sim\mathcal{M}(n,n,n;p)$ to contain a Latin square is $p=\log(n)/n$. The scale is suggested by the obstruction consisting of a line containing no $1$s: such a line appears around this probability, and the conjecture asserts that this is essentially the only obstruction to the existence of a Latin square. The paper does not establish the conjectured threshold; its preceding theorem only gives an infinite family of dimensions for which a fixed $p<1$ suffices.