Shin–Templier's negligible-poles conjecture for functorial L-functions

Let $\rho$ be an $L$-homomorphism, and let $\mathcal{P}$ be the characteristic function on $\hat{\mathbb{X}}_{\mathrm{gen}}$ of the representations $\pi$ for which $\Lambda(s,\rho_*\pi)$ has a pole in the critical strip. Let $J_X(\pi)$ be the cutoff function, $\ell(\pi)$ the harmonic weight, and $d\mu_{\mathrm{aut}}$ the automorphic Plancherel measure. Negligible-poles conjecture. There exists some $\eta(\rho)>0$ such that

$$\int_{\hat{\mathbb{X}}_{\mathrm{gen}}}\mathcal{P}(\pi)\frac{J_X(\pi)}{\ell(\pi)}d\mu_{\mathrm{aut}}(\pi)\ll X^{r-1-\eta(\rho)}.$$

This weighted estimate would imply that representations whose functorial $L$-functions have poles in the critical strip are negligible in the family; the source presents it as a quantitative version of the assertion that almost all such $L$-functions are entire.