Suppose $G$ is a semisimple algebraic group over the rational numbers, $\pi$ is a cuspidal automorphic representation of $G$, and $r: \widehat{G}(\mathbf{C}) \rightarrow \mathrm{GL}_N(\mathbf{C})$ is an irreducible representation of the Langlands dual group. Suppose moreover that $G$ is split at every finite place, and $\pi$ is unramified at every finite place. Then the $L$-function $L(\pi,r,s)$ is unambiguously defined. Is it conjectured that this $L$-function is non-vanishing at $s=1$?
Note that if $G = \mathrm{PGL}_n$ and $r$ is the standard representation, then the nonvanishing is known.