Non-vanishing conjecture for twisted central critical values

Let $F/F^+$ be the CM extension in the paper, let $S$ be a finite set of non-archimedean places of $F^+$, and for each $v\in S$ let $\alpha_v:\operatorname{GL}_1(\mathcal O_{F^+,v})\to\mathbb C^\times$ be a continuous character. Let $\chi_\infty$ be an algebraic character of $\operatorname{GL}_1(F\otimes_{\mathbb Q}\mathbb R)$. For the automorphic representation $\Pi$ under consideration, an algebraic Hecke character $\chi$ has the prescribed local restrictions and archimedean component.

Non-vanishing conjecture. There is an algebraic Hecke character $\chi$ of $\operatorname{GL}_1(\mathbb A_F)$, with conjugate self-dual archimedean component $\chi_\infty$, such that

$$\chi_{|_{\operatorname{GL}_1(\mathcal O_{F^+,v})}}=\alpha_v$$

for every $v\in S$, and

$$L(\tfrac12,\Pi\otimes\chi)\ne0.$$

The conjecture supplies the non-vanishing twisted central value needed to remove a regularity hypothesis in the paper’s rationality arguments. The source gives no resolution.