Alon–Tarsi conjecture on the parity of Latin squares

Let $L$ be a Latin square of order $n$, and define its sign by the product of the signs of its row and column permutations. Let $E_n$ and $O_n$ denote the numbers of even and odd Latin squares, respectively. Alon–Tarsi conjecture. If $n$ is even, then

$$E_n-O_n\neq 0,$$

equivalently, the signed enumeration of Latin squares is nonzero. The conjecture remains open in general, with nontrivial cases known for several infinite families of even orders.