Sheats's order-independence conjecture for symplectic jeu de taquin

Let $T$ be an admissible skew tableau. At each step, choose an inside corner and apply the symplectic jeu de taquin sliding procedure, iterating until a symplectic tableau is obtained. Sheats's conjecture. The resulting symplectic tableau is independent of the order in which the inner corners are filled. This asserts a confluence property for the symplectic jeu de taquin algorithm, extending the order-independence familiar from ordinary jeu de taquin; the supplied text does not state whether the conjecture has been proved or disproved.