Nonvanishing of central Rankin–Selberg $L$-values after twisting

Let $\Pi$ and $\Pi'$ be the automorphic representations specified in the surrounding hypotheses, and let $\Pi_\chi$ and $\Pi_{\chi'}$ be the associated twists by conjugate self-dual Hecke characters $\chi$ and $\chi'$. For a finite set of places $S$, write $L^S(s,{}^\sigma\Pi_\chi\times{}^\sigma\Pi_{\chi'})$ for the corresponding partial Rankin–Selberg $L$-function. Nonvanishing conjecture. There are characters $\chi$ and $\chi'$ such that

$$L^S\left(\tfrac12,{}^\sigma\Pi_\chi\times{}^\sigma\Pi_{\chi'}\right)\neq 0$$

for all $\sigma\in\operatorname{Aut}(\mathbb C)$. The statement predicts simultaneous nonvanishing under every field automorphism of $\mathbb C$; the supplied context gives no resolution status.