Functorial lifting conjecture from $G_2$ to $GL_7$

Let $\Pi$ be a cuspidal automorphic representation of $G_2(\mathbb A)$ whose archimedean component is cohomological of regular weight

$$\lambda=c_1(\alpha+2\beta)+c_2(2\alpha+3\beta),\qquad c_1,c_2>0.$$

Let $R_7$ be the standard $7$-dimensional representation of $G_2$.

Functorial lifting conjecture. There is a self-dual automorphic representation $\widetilde\Pi$ of $GL_7(\mathbb A)$ satisfying all five listed properties: matching unramified Satake parameters via $R_7$; cohomological archimedean weight $(2c_1+c_2,c_1+c_2,c_1,0,-c_1,-c_1-c_2,-2c_1-c_2)$; an isobaric decomposition into cuspidal representations on $GL_{n_i}$ with $\sum n_i=7$; and the stated Langlands-quotient descriptions at finite places associated respectively with the parabolics $P_{2,3,2}$ and $P_{1,2,1,2,1}$. This conjecture predicts the expected standard functorial transfer from $G_2$ to $GL_7$, including compatibility with local parabolic data. The source uses it in the study of cuspidal members of a $p$-adic deformation and does not state that it has been proved.