Exceptional-pole conjecture for GSp(4) × GL(2) L-factors

Let $\pi$ and $\sigma$ be generic irreducible representations of $\operatorname{GSp}(4,F)$ and $\operatorname{GL}(2,F)$, with central characters $\chi_{\pi}$ and $\chi_{\sigma}$. Let $s_0\in\mathbf C$ satisfy

$$\chi_{\pi}\chi_{\sigma}|\cdot|^{2s_0}=1.$$

An exceptional pole is a pole in the exceptional part of Novodvorsky's local $L$-factor.

Exceptional-pole conjecture. The point $s_0$ is an exceptional pole of $L(\pi\times\sigma,s)$ if and only if it is a pole of

$$\frac{L(\pi\times\sigma,s)L(\pi\times\sigma,s+1)}{L(\pi\times\sigma\times\operatorname{St},s+\tfrac12)}.$$

Equivalently, the $8$-dimensional Weil–Deligne representation $\phi_{\pi}\otimes\phi_{\sigma}$ has a $1$-dimensional unramified direct summand whose $L$-factor has a pole at $s_0$. The conjecture gives a Langlands-parameter interpretation of exceptional poles; its general validity is not established in the source.