Uniqueness of terminal transversals under $(213)\rightarrow(123)$ moves

Let $Y$ be the Young diagram under consideration, let $S_Y$ denote its transversals, and let $S_Y(213)$ denote the transversals avoiding the pattern $(213)$. For a transversal $T\in S_Y$, consider sequences of $(213)\rightarrow(123)$ moves, each replacing an occurrence of the pattern $(213)$ by $(123)$. Uniqueness conjecture. Starting with a transversal $T\in S_Y$, all sequences of $(213)\rightarrow(123)$ moves terminate in the same transversal $T'\in S_Y(213)$. This is a confluence assertion for the pattern moves and would make the $(213)$-avoiding terminal transversal canonically determined by the starting transversal. The supplied text does not indicate whether the assertion has been proved or remains open.