Cuspidal occurrence conjecture for the long-root Langlands quotient of $G_2$

Let $F$ be a cuspidal holomorphic eigenform of weight $k\geq4$ and trivial nebentypus, let $\pi_F$ be its associated automorphic representation, and set $\Pi_{F,f}=\mathcal L_\alpha(\pi_F,1/10)_f$, the finite part of the Langlands quotient from the long-root parabolic in $G_2$. Let $v$ be a place where $\pi_F$ is unramified, and let $\Pi_{F,f}^v$ be the component away from $v$. Assume $L(1/2,\pi_F,\operatorname{Sym}^3)=0$.

Cuspidal occurrence conjecture. The $\Pi_{F,f}^v$-isotypic component of the cuspidal spectrum is

$$L_{\operatorname{cusp}}^2(G_2(\mathbb Q)\backslash G_2(\mathbb A))[\Pi_{F,f}^v]=\Pi_{F,f}\otimes\Pi_\infty,$$

where $\Pi_\infty=\mathcal L_\alpha(\pi_F,1/10)_\infty$ if $\epsilon(1/2,\pi_F,\operatorname{Sym}^3)=1$, while if the root number is $-1$, $\Pi_\infty$ is the quaternionic discrete series of $G_2(\mathbb R)$ with Harish-Chandra parameter $\tfrac{k-4}{2}(2\alpha+3\beta)+\rho$. The conjecture is motivated by Arthur's multiplicity formula and predicts precisely which archimedean member occurs in the cuspidal spectrum; the source presents it as an unproved assertion.