Multiplicity and archimedean packet conjecture for $G_2$ lifts from $\mathrm{GL}_2$

Let $\pi$ be a unitary cuspidal automorphic representation of $\mathrm{GL}_2(\mathbb{A})$. Write

$$\pi=\bigotimes_v'\pi_v.$$

For each place $v$, let $\mathcal{L}_\alpha(\pi_v,1/10)$ denote the Langlands quotient of the unitary induction of $\pi_v\otimes|\operatorname{det}|^{1/2}$ from $M_\alpha(\mathbb{Q}_v)$ to $\mathrm{G}_2(\mathbb{Q}_v)$. Let $S$ be the set of places $v$ for which $\pi_v$ is discrete series.

Multiplicity and packet conjecture. For every $v\in S$, there is a representation $\Pi_v^-$ of $\mathrm{G}_2(\mathbb{Q}_v)$, different from $\mathcal{L}_\alpha(\pi_v,1/10)$, such that, for every subset $S'\subset S$,

$$\Pi=\bigotimes_{v\in S'}\Pi_v^-\otimes\sideset{}{'}\bigotimes_{v\notin S'}\mathcal{L}_\alpha(\pi_v,1/10)$$

occurs in $L^2_{\operatorname{disc}}(\mathrm{G}_2(\mathbb{Q})\backslash\mathrm{G}_2(\mathbb{A}))$ with multiplicity zero or one, and it has multiplicity one if and only if either $\epsilon(\pi,\operatorname{Sym}^3,1/2)=1$ and $\#S$ is even, or $\epsilon(\pi,\operatorname{Sym}^3,1/2)=-1$ and $\#S$ is odd. If $L(\pi,\operatorname{Sym}^3,1/2)=0$, then the representations $\Pi$ above that occur in the discrete spectrum are cuspidal. If $\pi_\infty$ is the discrete series of $\mathrm{GL}_2(\mathbb{R})$ of even weight $k\geq4$, then $\Pi_\infty^-$ is the discrete series representation of $\mathrm{G}_2(\mathbb{R})$ with Harish-Chandra parameter

$$\frac{k-4}{2}(2\alpha+3\beta)+\rho.$$

This conjecture predicts the discrete-spectrum multiplicities and the additional local member of the $G_2$ packet associated with a symmetric-cube lift from $\mathrm{GL}_2$. It also predicts cuspidality under the stated central-value vanishing condition and identifies the archimedean member in the even-weight discrete-series case.