Parity conjecture for type-B stack-sorting fibers

Let $B_n$ be the type-$B$ Coxeter group and let $\operatorname{\mathtt{s}}_B:B_n\to B_n$ be the type-$B$ stack-sorting map. For $v\in B_n$, write $\operatorname{\mathtt{s}}_B^{-1}(v)$ for its fiber. Parity conjecture. If $n\geq 1$ is odd and $v\in B_n$, then

$$\lvert \operatorname{\mathtt{s}}_B^{-1}(v)\rvert$$

is even. This conjecture predicts hidden parity structure in the fibers of the type-$B$ stack-sorting map; the source presents it as an unproved conjecture, and its status is therefore open.