Property of inner tableau translation for the weak order

Let $SYT_n$ denote the set of standard Young tableaux with $n$ cells, and let $]_{weak}$ be Melnikov's weak order on $SYT_n$. For tableaux $S,T$ with the same inner tableau $R$, let $S'$ and $T'$ be obtained by replacing $R$ in $S$ and $T$ by another tableau $R'$ of the same shape.

Property of inner tableau translation. If

$$S\mathrel{<_{weak}}T,$$

then

$$S'\mathrel{<_{weak}}T'.$$

This property is known for the Kazhdan–Lusztig and geometric orders; for the weak order, it would provide a self-contained proof that the order is well defined. The paper proves the conjecture in special cases, while the general statement remains unresolved.