The shifted Coxeter–Knuth insertion conjecture for involution words

Let an involution word be a word in the generators of the relevant involution-Hecke monoid, and let shifted Coxeter–Knuth equivalence be the equivalence relation generated by the four relations in Definition~. Write $P(\textbf{a})$ for the insertion tableau of an involution word $\textbf{a}$ under the map $SH$. Shifted Coxeter–Knuth insertion conjecture. Two involution words $\textbf{a}$ and $\textbf{b}$ are shifted Coxeter–Knuth equivalent if and only if

$$P(\textbf{a})=P(\textbf{b}).$$

This conjecture is the analogue of the corresponding characterizations for ordinary Coxeter–Knuth equivalence and shifted tableaux; its status is not resolved in the supplied source context.