Odd-cycle two-line trade conjecture for Latin squares

Let $\mathcal{L}_n$ be the set of Latin squares of even order $n$. In each of the row, column, and symbol views, a two-line trade is obtained by swapping two lines on the support of a cycle of the induced permutation; an odd-cycle trade is one whose cycle has odd length. Odd-cycle two-line trade conjecture. Every $L\in\mathcal{L}_n$ admits an odd-cycle two-line trade in at least one of the three conjugate views, and such a trade can be found among pairs belonging to a fixed canonical 1-factorization of the lines in that view. The claim is motivated by experiments and would provide systematic sign-flipping trades, but no proof is supplied.