Level-one exceptional theta-lift conjecture for automorphic representations of F4

Let $n$ be such that there is a cuspidal Hecke eigenform of weight $2n+12$ for $\operatorname{SL}_{2}(\mathbb{Z})$, and let $\pi$ be the associated level-one algebraic automorphic representation of $\operatorname{PGL}_{2}$. Let $c_p$ be the Satake parameter of $\pi_p$, viewed as a semisimple conjugacy class in $\widehat{\operatorname{PGL}_{2}}(\mathbb{C})=\operatorname{SL}_{2}(\mathbb{C})$. Let $\varpi_4$ be the highest weight of the $26$-dimensional irreducible representation of $\mathbf{F}_{4}(\mathbb{R})$, and let $\iota:\operatorname{SL}_{2}(\mathbb{C})\times\operatorname{SL}_{2}(\mathbb{C})\to\mathbf{F}_{4}(\mathbb{C})$ be the morphism obtained from the principal embedding in $\operatorname{Sp}_{6}(\mathbb{C})$ and the identity in the second factor. For each prime $p$, let $e_p$ be the conjugacy class of $\left(\begin{smallmatrix}p^{1/2}&0\\0&p^{-1/2}\end{smallmatrix}\right)$ in $\operatorname{SL}_{2}(\mathbb{C})$. Level-one exceptional theta-lift conjecture. There exists a level-one automorphic representation $\Pi$ of $\mathbf{F}_{4}$ such that

$$\Pi_{\infty}\simeq \mathrm{V}_{n\varpi_{4}},$$

where $\mathrm{V}_{n\varpi_{4}}$ is the irreducible representation of $\mathbf{F}_{4}(\mathbb{R})$ with highest weight $n\varpi_{4}$, and, for every prime $p$, the Satake parameter of $\Pi_p$ is the conjugacy class of $\iota(e_p,c_p)$. This conjecture predicts a specific level-one family of automorphic representations of the exceptional group $\mathbf{F}_{4}$ attached to classical modular forms; the supplied text gives no resolution status.