Nonvanishing conjecture for Rankin–Selberg central values

Let ${\bf{f}} \in \mathcal{S}_2(\mathfrak{N})$ be a cuspidal Hilbert modular eigenform, and let $K/F$ be a totally imaginary quadratic extension such that the root number of $L({\bf{f}},K,s)$ is $+1$. For each positive integer $j$, let $\mathfrak{S}_j^+$ be the set of admissible squarefree ideals defined in the paper for which the relevant root number is $+1$, and let ${\bf{f}}^{(\mathfrak{n})}$ denote the corresponding congruent eigenform. Nonvanishing conjecture. There exists a positive integer $k_0$ such that, for every integer $j\geq k_0$, some ideal $\mathfrak{n}\in\mathfrak{S}_j^+$ satisfies

$$L({\bf{f}}^{(\mathfrak{n})},K,1)\ne 0.$$

This is the reformulation of Howard's criterion at the augmentation prime in terms of complex Rankin–Selberg central values. It is proposed as the final nonvanishing statement in the paper and is not established there.