The weak epsilon dichotomy conjecture for the GSp4 times GL2 times GL2 model

Let $G=\operatorname{GSp}_4\times\operatorname{GL}_2\times\operatorname{GL}_2$ with $H=(\operatorname{GL}_2\times\operatorname{GL}_2)^0$, let $(G_D,H_D)$ be its quaternionic inner form, and let $\rho_X$ be the tensor product of the two standard $\operatorname{GL}_2(\mathbb C)$ representations with the Spin representation of $\operatorname{GSpin}_5(\mathbb C)$. Let $\Pi_\phi$ be a tempered $L$-packet with central character trivial on $Z_{G,H}(F)$. The weak epsilon dichotomy conjecture for the GSp4 times GL2 times GL2 model. The unique distinguished element lies in $\Pi_\phi(G)$ exactly when $\epsilon(1/2,\Pi_\phi,\rho_X)=1$, and lies in $\Pi_\phi(G_D)$ exactly when $\epsilon(1/2,\Pi_\phi,\rho_X)=-1$. This is the weak epsilon dichotomy prediction for the model; the source supplies no resolution evidence.