Geometric epsilon-factor conjecture for Shalika periods

Let $F$ be non-archimedean, let $E/F$ be quadratic, and let $\mathbf{G}$ and $\mathbf{H}$ be the groups in the twisted linear-period setting. Let $\pi'\in\operatorname{Irr}_\mathrm{disc}(\mathbf{G}(F))$ have central character $\omega=(\chi\circ\operatorname{N}_{E/F})^{n/2}$, satisfy $\pi'\cong\pi'^c$, and have Langlands parameter valued in $\mathrm{GSp}_n(\mathbb{C})$ with similitude factor $\chi\circ\operatorname{N}_{E/F}$. Define

$$\varepsilon(\pi',\chi)=\varepsilon\left(\frac12,\pi'\otimes\chi^{-1},\psi_E\right)\chi(-1)^{\frac n2}.$$

Geometric epsilon-factor conjecture. Under these hypotheses,

$$\varepsilon_\mathrm{geom}(\pi',\chi)=\varepsilon(\pi',\chi).$$

The conjecture identifies a geometric expression from twisted character germs with the standard epsilon factor; no resolution status is supplied in the text.