Johansson's Latin-square threshold conjecture

For each $n$, independently include each symbol in each list $L_{ij}\subseteq\{1,\dots,n\}$ with probability $p$, and let $\mathcal{F}_{\mathrm{Latin}}$ be the property that there is a Latin square whose entry in position $(i,j)$ belongs to $L_{ij}$. Johansson's Latin-square threshold conjecture. The property $\mathcal{F}_{\mathrm{Latin}}$ has a sharp threshold at

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

The scale is forced by the local obstruction of empty lists; the conjecture also concerns the corresponding asymptotic, first-order, and hitting-time behavior, while the supplied statement records only sharpness at this scale.