Cameron's conjecture on row parities in random Latin squares

Let $\mathbf{L}$ be a uniformly random $n\times n$ Latin square, and let $N_{\mathrm{row}}(\mathbf{L})$ denote the number of odd row permutations of $\mathbf{L}$. Cameron's conjecture. As $n\to\infty$, the distribution of $N_{\mathrm{row}}(\mathbf{L})$ is approximately the binomial distribution $\operatorname{Bin}(n,1/2)$. This conjecture formalizes the idea that the row parities of a random Latin square behave like $n$ independent fair coin flips. The Alon--Tarsi conjecture would then hold only just barely, while the precise meaning and validity of this asymptotic approximation remain open.