Birational rowvacuation homomesy for type B quotient posets

Let $\mathsf{B}^n$ be the type B quotient poset, let $\mathcal{A}^{\mathrm{B}}_{\kappa}(\mathsf{B}^n)$ denote its $\kappa$-normalized birational labelings, and let $h^{\mathrm{B}}_i$ for $1\leq i\leq 2n-1$ and $h^{\mathrm{B}}_*$ be the birational statistics defined by detropicalizing the corresponding piecewise-linear statistics.

Birational type B homomesy conjecture. The statistics $h^{\mathrm{B}}_i\colon \mathcal{A}^{\mathrm{B}}_{\kappa}(\mathsf{B}^n)\to \mathbb{R}_{>0}$, for $1\leq i\leq 2n-1$, and $h^{\mathrm{B}}_*\colon \mathcal{A}^{\mathrm{B}}_{\kappa}(\mathsf{B}^n)\to \mathbb{R}_{>0}$, are multiplicatively $\kappa$-mesic for $\operatorname{Rvac}_{\mathcal{A}}^{\mathrm{B}}$. The analogous statement for the type $\mathsf{B}'^n$ quotient is proved using an embedding, whereas the type $\mathsf{B}^n$ case is conjectured because that technique is obstructed by factors of $2$.