A barred signed Edelman–Greene expansion at opposite alphabets

Let $C_{n+1}$ be the hyperoctahedral group, let $\mathbf{x}$ be an alphabet of variables, and let $F_{\omega}^d$ denote the doubled Stanley symmetric function. Suppose $\omega\in C_{n+1}$ is unknotted and every reduced word for $\omega$ contains at most one $s_0$. Let $\bar{E}_{\omega}^{\lambda r}$ be the number of signed Edelman–Greene tableaux for $\omega$ of shape $\lambda$ with exactly $r$ barred entries.

Barred signed Edelman–Greene expansion. Then

$$F_{\omega}^{d}(\mathbf{x},-\mathbf{x})=\sum_{r\text{ even}}\sum_{\lambda}\bar{E}_{\omega}^{\lambda r}s_{\lambda}(\mathbf{x})-\sum_{r\text{ odd}}\sum_{\lambda}\bar{E}_{\omega}^{\lambda r}s_{\lambda}(\mathbf{x}).$$

This is the second of three conjectures in the section; the provided text gives no resolution, while the stronger untangled case is stated to be a theorem.