Residual-bias involution principle for the Alon–Tarsi conjecture

Let $\mathcal{L}_n$ be the set of Latin squares of even order $n$, let $\operatorname{sgn}(L)$ be the standard Alon–Tarsi sign, and let $\operatorname{Res}_n$ be a residual set. Residual-bias involution principle. For each even $n$, there exists a deterministic scan order $\mathcal{S}_n$ of two-line cycle trades and a deterministic stability rule such that the induced map $F_n:\mathcal{L}_n\to\mathcal{L}_n$ is an involution and

$$\operatorname{sgn}(F_n(L))=-\operatorname{sgn}(L)$$

for every $L\in\mathcal{L}_n\setminus\operatorname{Res}_n$, where $\operatorname{Res}_n$ is nonempty and has nonzero signed sum, for instance $\sum_{L\in\operatorname{Res}_n}\operatorname{sgn}(L)=\pm|\operatorname{Res}_n|$. Consequently, the signed enumeration of Latin squares is nonzero. This would prove the Alon–Tarsi conjecture at every even order, but the supplied text gives only an experimentally suggested principle and no proof.