Orthogonal Sagan–Worley insertion conjecture for orthogonal Knuth operators

Let $a=a_1a_2\cdots a_n$ be a primed word, let $i\in\mathbb{Z}$, and let $\mathsf{okn}_i$ be the orthogonal Knuth operator defined on primed words. Let $P^\mathsf{O}_{\textsf{SW}}(a)$ and $Q^\mathsf{O}_{\textsf{SW}}(a)$ denote the insertion and recording tableaux under orthogonal Sagan–Worley insertion, and let $\mathfrak{d}_i$ denote the corresponding dual-equivalence operation on recording tableaux.

Orthogonal Sagan–Worley insertion conjecture. For every $i\in\mathbb{Z}$,

$$P^\mathsf{O}_{\textsf{SW}}(\mathsf{okn}_i(a))=P^\mathsf{O}_{\textsf{SW}}(a)$$

and

$$Q^\mathsf{O}_{\textsf{SW}}(\mathsf{okn}_i(a))=\mathfrak{d}_i(Q^\mathsf{O}_{\textsf{SW}}(a)).$$

These identities are immediate for $i\in\{-1,0\}$, while the case $1\leq i\in\ell(a)-2$ is the difficult one. If true, the conjecture would imply that two primed words are related by the transitive closure of the orthogonal Knuth relations exactly when they have the same orthogonal Sagan–Worley insertion tableau.