Beyond endoscopy pole detection conjecture for functorial transfers

Let $F$ be a number field, let $\mathbb{A}_F$ be its ring of adeles, let $G$ be a reductive group over $F$, and let ${}^{\lambda}G$ be the Zariski closure of the image of ${}^{L}G$ viewed as a reductive group over $\mathbb{C}$. Let $r:\mathrm{GL}_n\longrightarrow\mathrm{GL}(V_r)$ be a representation that detects ${}^{\lambda}G$, meaning that ${}^{\lambda}G$ fixes a line in $V_r$. Let $\pi$ be a unitary cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_F)$, and let $\chi$ be a character of $F^{\times}\backslash\mathbb{A}_F^{\times}$. Beyond endoscopy pole detection conjecture. If $\pi$ is a functorial transfer from $G$, then $L(s,\pi,r\otimes\chi)$ has a pole at $s=1$ for some character

$$\chi\in F^{\times}\backslash\mathbb{A}_F^{\times}\to\mathbb{C}^{\times}$$

whenever $r$ detects ${}^{\lambda}G$. This conjecture gives a proposed analytic characterization of functorial transfers, forming part of Langlands' beyond endoscopy program. In the case covered by the source, the conjecture is proven by work of Arthur, Cogdell, Kim, Piatetski-Shapiro, Shahidi, Ginzburg, Rallis, and Soundry; the general beyond endoscopy program remains a broader source of open problems.