Cover-preserving inner tableau translation conjecture for the weak order

For $1\leq k<n$ and $R\in SYT_k$, define

$$SYT_n^R:=\{T\in SYT_n\mid T_{[1,k]}=R\}.$$

For tableaux $R,R'\in SYT_k$ of the same shape, let

$$\mathcal{V}_{[R,R']}:SYT_n^R\mapsto SYT_n^{R'}$$

be the inner tableau translation map obtained by replacing $R$ with $R'$. A covering relation $S\lessdot_{weak}T$ in $SYT_n^R$ is translated to the corresponding tableaux in $SYT_n^{R'}$.

Inner tableau translation conjecture. If $S\lessdot_{weak}T$ is a covering relation in $SYT_n^R$ and $R'$ is obtained from $R$ by applying a single dual Knuth relation, then

$$\mathcal{V}_{[R,R']}(S)\lessdot_{weak}\mathcal{V}_{[R,R']}(T)\quad\text{in }SYT_n^{R'}.$$

Equivalently, the weak order on standard Young tableaux is preserved under the inner tableau translation map.

This is a weaker formulation of the general inner tableau translation property. The corresponding preservation statement is known for the Kazhdan–Lusztig and geometric orders, whereas the weak-order conjecture is posed here as an open question; the paper establishes special cases.