Local-global symmetric power functoriality conjecture for essentially discrete series

Let $n\geq 1$. Let $F^+$ be a totally real field, and let $\pi$ be a cuspidal automorphic representation of $\mathrm{GL}_2({\mathbf A}_{F^+})$ without CM, with $\pi_\infty$ essentially square-integrable. Local-global symmetric power functoriality conjecture. There exists a cuspidal automorphic representation $\Pi$ of $\mathrm{GL}_n({\mathbf A}_{F^+})$ such that, for every place $v$ of $F^+$,

$$\operatorname{rec}_{F^+_v}(\Pi_v)\cong\operatorname{Sym}^{n-1}\circ\operatorname{rec}_{F^+_v}(\pi_v).$$

This extends the preceding symmetric-power formulation beyond RAESDC representations by imposing an essentially square-integrable condition at infinity. It asserts compatibility at every local place, and no resolution is given here.