Orthogonal Knuth equivalence characterization by Sagan–Worley insertion

Let $a$ and $b$ be primed words, and let $\overset{\textsf{shK}}\sim$ be the transitive closure of the relation $a\sim\mathsf{okn}_i(a)$ for all $i\in\mathbb{Z}$. Let $P^\mathsf{O}_{\textsf{SW}}(a)$ denote the insertion tableau under orthogonal Sagan–Worley insertion.

Orthogonal Knuth equivalence conjecture. Two primed words satisfy

$$a\overset{\textsf{shK}}\sim b$$

if and only if

$$P^\mathsf{O}_{\textsf{SW}}(a)=P^\mathsf{O}_{\textsf{SW}}(b).$$

This is presented as a consequence of the orthogonal Sagan–Worley insertion conjecture. The paper establishes that every primed word is orthogonally Knuth equivalent to the row word of its insertion tableau, but the converse characterization remains conjectural.