Chai's conjecture on exceptional poles of archimedean Rankin–Selberg L-functions

Let $F$ be a local field of characteristic $0$, let $pi_1$ and $pi_2$ be irreducible generic representations of $\operatorname{GL}_n(F)$, and let $m\geq 0$. Exceptional poles of type $1$ with level $m$ are defined through the Rankin–Selberg integrals, while exceptional poles of type $2$ with level $m$ are defined through the corresponding $\eta$-homology and representation-theoretic data. In particular, a type-$2$ pole at $s=s_0$ satisfies

$$\operatorname{Hom}_{G_n}\!\bigl(\pi_1\hat{\otimes}\pi_2\hat{\otimes}\operatorname{Sym}^m(\mathbb{C}^n),\;\left|\operatorname{det}\right|^{-s_0}\bigr)\neq 0.$$

Chai's conjecture. A complex number $s=s_0$ is an exceptional pole of type $1$ with level $m$ for the pair $(\pi_1,\pi_2)$ if and only if it is an exceptional pole of type $2$ with level $m$. This conjecture predicts that the analytic definition via Rankin–Selberg integrals and the representation-theoretic definition via $\eta$-homology give the same level-by-level notion of exceptional pole. The archimedean theory is less understood than the non-archimedean one, and the source provides no resolution of the conjecture.