Density threshold conjecture for partial Latin squares in random Latin squares

For a non-empty partial Latin square $P$, define its density by

$$m(P):=\min\left\{\frac{|\mathcal{R}_{P'}|+|\mathcal{C}_{P'}|+|\mathcal{S}_{P'}|-|P'|}{|P'|}:\varnothing\ne P'\subseteq P\right\}.$$

Let $\mathbf{L}$ be a random Latin square of order $n$. The density threshold conjecture. As $n\to\infty$,

$$\begin{cases} \mathop{\mathbb{P}}\nolimits(\mathbf{L}\text{ contains an isotopic copy of }P)\to 0 & \text{if }m(P)<0,\\ \mathop{\mathbb{P}}\nolimits(\mathbf{L}\text{ contains an isotopic copy of }P)\to 1 & \text{if }m(P)>0. \end{cases}$$

When $m(P)=0$, the source additionally expects the number of copies of $P$ in $\mathbf{L}$ to have a Poisson distribution, mentioning $3\times3$ subsquares as an example. The threshold dichotomy itself remains conjectural in the supplied text.