The involution-word bumping conjecture for queer supercrystals

Let $I_\mathbb{Z}$ denote the set of involutions in the infinite symmetric group, let $w=w_1w_2\cdots w_n$ be an involution word, and let $\mathfrak{ib}_\pi$ be the involution-word bumping operator. Suppose

$$\mathfrak{ib}_\pi(w)=\tilde w_1\tilde w_2\cdots\tilde w_n.$$

Involution-word bumping conjecture. For every $i\in[n]$, one has $\tilde w_i-w_i\in\{0,1\}$.

Computations support this analogue of the corresponding result for Edelman–Greene insertion; if true, it would immediately prove the most difficult part of the stated theorem on the orthogonal insertion algorithm. The conjecture is presented as open in the source.