On the threshold problem for Latin boxes

Let $n \leq m \leq k$. An $n \times m \times k$ 0-1 array is a Latin box if it contains exactly $nm$ ones, and has at most one $1$ in each line. Let $\mathcal{M}(n,m,k;p)$ be the distribution on $n \times m \times k$ 0-1 arrays where each entry is $1$ with probability $p$, independently of the other entries. For which values of $p$ does $M \sim \mathcal{M}(n,m,k;p)$ support a Latin box with high probability? As Latin squares are equivalent to $n \times n \times n$ Latin boxes, this question is a generalization of the threshold question for Latin squares. Let $\varepsilon>0$. We give an asymptotically tight answer to this question in the special cases where $m=k$ and $n \leq \left(1-\varepsilon \right) m$ and where $n=m$ and $k \geq \left(1+\varepsilon \right) n$. In both cases, the threshold probability is $\Theta \left( \log \left( n \right) / n \right)$. This implies threshold results for Latin rectangles and proper edge colorings of $K_{n,n}$.