Concatenation conjecture for standard Young tableaux and vacillating tableaux

Let standard Young tableaux be concatenated when their row lengths all have the same parity, and let vacillating tableaux of shape $\emptyset$ be concatenated in the corresponding way. Concatenation conjecture. Concatenation of standard Young tableaux whose row lengths all have the same parity corresponds, in general, to concatenation of vacillating tableaux of shape $\emptyset$. This is proved for $k=1$ and for standard Young tableaux with even row length; the general case remains open in the source.