Nonvanishing conjecture for the base-change trace

Let $F$ and $E$ be number fields as in the paper, let $n$ be a positive integer, and let $_{E/F}(\Sigma_{\phi}^{S_0}(X))$ denote the associated base-change test function. For a cuspidal automorphic representation $\pi$ of $A_{\mathrm{GL}_{nF}} \backslash \mathrm{GL}_n(\mathbb{A}_F)$, nonvanishing conjecture.

$$\lim_{X \to \infty}X^{-1}\mathrm{tr}(\pi)(b_{E/F}(\Sigma_{\phi}^{S_0}(X))) \neq 0.$$

This nonvanishing would supply the missing hypothesis in the weak converse theorem and would imply existence of weak base change for every cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_F)$.