The Lascoux-Schützenberger involutions, $s_i$, that permute the weight of semi-standard Young tableaux are fairly known.

They satisfy some nice Coxeter relations, for example, if $v$ and $w$ are reduced words of the same permutation, then $s_{v_1} \dots s_{v_l} = s_{w_1} \dots s_{w_l}$.

Each $s_i$ only acts on the entries $i$ and $i+1$ in the tableau, and columns that contain one entry of both $i$ and $i+1$ is fixed.

Now, is there some generalization of these involutions that

- Generalize to plane partitions (or arbitrary fillings), where the same element can appear more than once in both rows and columns,
- satisfy the independence of reduced word relation
- specialize to the classical involutions on semi-standard Young tableaux?

Note that this generalization does not need to preserve the position of other entries if the filling is not a SSYT: What I am looking for will *not* have this property.