Shifted Hecke insertion conjecture for involution words

Let $v$ and $w$ be involution words, and let $\overset{\textsf{O}}\sim$ denote the shifted Hecke equivalence relation. The shifted Hecke insertion algorithm assigns to each word $u$ a tableau $P_{\textsf{O}}(u)$. Shifted Hecke insertion conjecture. If

$$v\overset{\textsf{O}}\sim w,$$

then

$$P_{\textsf{O}}(v)=P_{\textsf{O}}(w).$$

The claim concerns the converse to the already established implication from equal insertion tableaux to equivalence. It is supported by computations and is also stated as Conjecture 5.13 in work of Hamaker, Marberg, and Pawlowski.