A signed Edelman–Greene expansion for doubled Stanley symmetric functions

Let $C_{n+1}$ be the hyperoctahedral group, and let $\boldsymbol{x}$ be an alphabet of variables. For an unknotted $\omega\in C_{n+1}$, write $F_{\omega}^d$ for its doubled Stanley symmetric function. A signed Edelman–Greene tableau for $\omega$ is a tableau whose entries, reading word, and shape are as defined in the source; let $\bar{E}_{\omega}^{\lambda}$ denote the number of such tableaux of shape $\lambda$.

Signed Edelman–Greene expansion. If $\omega\in C_{n+1}$ is unknotted, then

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

The analogous formulas are theorems for the stronger condition of untangledness, while the assertion for all unknotted elements is presented as a conjecture.