The weak epsilon dichotomy conjecture for smaller GSpin and GHSpin models

Let $(G,H)$ be one of the smaller models $(\operatorname{GSpin}_{10}\times\operatorname{GSpin}_3,\operatorname{GSpin}_3\ltimes U)$, $(\operatorname{GHSpin}_{12},\operatorname{PGL}_2\ltimes U)$, or $(\operatorname{GHSpin}_{12}\times\operatorname{GL}_2,\operatorname{GL}_2\ltimes U)$, with pure inner form $(G_D,H_D)$, and let $\rho_X$ be respectively the representation specified in the source. Let $\Pi_\phi=\Pi_\phi(G)\cup\Pi_\phi(G_D)$ be a tempered $L$-packet whose central character is trivial on $Z_{G,H}(F)$. The weak epsilon dichotomy conjecture for the smaller GSpin and GHSpin models. The unique distinguished element lies in $\Pi_\phi(G)$ exactly when

$$\epsilon\left(\frac12,\Pi_\phi,\rho_X\right)=1,$$

and lies in $\Pi_\phi(G_D)$ exactly when the same epsilon factor equals $-1$. The conjecture packages the predicted pure inner form for the distinguished representation; no resolution evidence is supplied.