The fixed-point-free involution bumping conjecture

Let $I^{\mathsf{FPF}}_\mathbb{Z}$ denote the set of fixed-point-free involutions in the infinite symmetric group, let $w=w_1w_2\cdots w_n$ be an fpf-involution word, and let $\mathfrak{fb}_\pi$ be the fixed-point-free involution bumping operator. Suppose

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

Fixed-point-free involution bumping conjecture. For every $i\in[n]$, one has $\tilde w_i-w_i\in\{0,1,2\}$.

This is described as a weaker version of the preceding conjecture for the $\mathfrak{fb}_\pi$ operators. The source notes that it does not yield a simple proof of the corresponding theorem because letters may be incremented twice, and presents the claim as open.