The weak epsilon dichotomy conjecture for the GL4 times GL2 model

Let $G=\operatorname{GL}_4\times\operatorname{GL}_2$, let $H=\operatorname{GL}_2\times\operatorname{GL}_2$ be embedded by $(a,b)\mapsto(\operatorname{diag}(a,b),b)$, and let $G_D=\operatorname{GL}_2(D)\times\operatorname{GL}_1(D)$ be the pure inner form, where $D$ is the quaternion algebra over $F$. Let $\rho_X$ be the representation specified in the paper, and let $m(\pi)$ denote the distinguished multiplicity. The weak epsilon dichotomy conjecture for the GL4 times GL2 model. For an irreducible tempered representation $\pi=\pi_1\otimes\pi_2$ of $G(F)$ whose central character is trivial on $Z_{G,H}(F)$, with Jacquet–Langlands transfer $\pi_D$ when it exists and $\pi_D=0$ otherwise,

$$m(\pi)=1\iff\epsilon(1/2,\pi,\rho_X)=1,$$ $$m(\pi_D)=1\iff\epsilon(1/2,\pi,\rho_X)=-1.$$

This is the explicit form of the paper's epsilon dichotomy prediction for this model; the source provides no resolution evidence.