Stable odd-cycle trade conjecture with bounded isotopy depth

Let $\mathcal{L}_n$ be the set of Latin squares of even order $n$. A stable odd-cycle two-line trade is an odd-cycle trade whose canonical selection remains compatible with reapplication, and let $P$ range over single transpositions of two rows or two columns. Stable odd-cycle trade conjecture. For every $L\in\mathcal{L}_n$, there exists a stable odd-cycle two-line trade in $L$ or in a conjugate $P(L)$ such that the stabilized canonical rule selects the same $(P,\text{trade})$ for $L$ and $F(L)$. Consequently, the resulting map is a sign-reversing involution except possibly on a residual set $\operatorname{Res}_n$ handled by a final canonical rule, with $\sum_{L\in\operatorname{Res}_n}\operatorname{sgn}(L)\neq 0$. The statement is an experimental proposal; the supplied text does not prove existence of the stable trades or the residual bias.