Horizontal non-vanishing conjecture for Hecke twists of automorphic Rankin–Selberg L-functions

Let $F$ be a totally real number field, $\mathbb{A}$ its ring of adeles, and $\pi$ a cuspidal cohomological automorphic representation of $\operatorname{GL}_{2}(\mathbb{A})$. For an ideal $\mathfrak{c}\subset\mathcal O_F$, a CM quadratic extension $K/F$, and an allowed infinity type $\kappa$, let $\mathfrak{X}_{K,\kappa,\mathfrak{c}}$ be the set of Hecke characters over $K$ satisfying (C1) and (C2). Let $i\in\{0,1\}$ satisfy $\epsilon(\pi,\chi_0)=(-1)^i$ for some $\chi_0\in\mathfrak{X}_{K,\kappa,\mathfrak{c}}$. Horizontal non-vanishing conjecture. For every $\epsilon>0$ and every $K\in\Theta_{\kappa,\mathfrak{c}}$, one has

\\#\left\\{\chi\in\mathfrak{X}_{K,\kappa,\mathfrak{c}} \\ \Big|\\ L^{(i)}(1/2,\pi,\chi)\neq0\right\\}\gg_{\epsilon,\pi}|D_K|^{\frac12-\epsilon}.

Moreover, for every infinite subset $\Theta\subset\Theta_{\kappa,\mathfrak{c}}$, the quantity on the left tends to infinity as $K$ ranges over $\Theta$. The conjecture predicts horizontal non-vanishing of central values, or central derivatives when the root number is $-1$, across CM extensions; the source further suggests that a positive-proportion non-vanishing statement may hold, but that stronger assertion is not included here as a separate row.